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

Matsubara, Takahiko; Kuriki, Satoshi

doi:10.48550/arxiv.2011.04954

“…In this section, we summarize the second-order formula of Minkowski functionals with weak non-Gaussianity derived in the previous papers [29,30].…”

Section: Analytic Formula Of Minkowski Functionals With Second-order ...mentioning

confidence: 99%

“…Analytic formula of the Minkowski functionals up to second order in weakly non-Gaussian field in general dimensions d is derived in Refs. [29,30]. In the case of three dimensions, d = 3, the derived formula reduces to…”

Section: Analytic Formula Of Minkowski Functionals With Second-order ...mentioning

confidence: 99%

“…Formal expression of Euler characteristic, or genus statistics, which is one of the Minkowski functionals, in two and three dimensions in terms of the Gram-Charlier expansion to all orders are known [27,28]. Most recently, an analytic formula for non-Gaussian corrections up to the second order are derived in general dimensions [29,30]. The second-order terms involve integrals of trispectrum, which are called kurtosis parameters.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: second-order effects

Matsubara¹,

Kuriki²

2020

Preprint

Self Cite

2

1

0

View full text Add to dashboard Cite

The second-order formula of Minkowski functionals in weakly non-Gaussian fields is compared with the numerical N-body simulations. Recently, weakly non-Gaussian formula of Minkowski functionals is extended to include the second-order effects of non-Gaussianity in general dimensions. We apply this formula to the three-dimensional density field in the large-scale structure of the Universe. The parameters of the secondorder formula include several kinds of skewness and kurtosis parameters. We apply the tree-level nonlinear perturbation theory to estimate these parameters. First we compare the theoretical values with those of numerical simulations on the basis of parameter values, and next we test the performance of the analytic formula combined with the perturbation theory. The second-order formula outperforms the first-order formula in general. The performance of the perturbation theory depends on the smoothing radius applied in defining the Minkowski functionals. The quantitative comparisons are presented in detail.

show abstract

“…As the field is mean-free, the third-order cumulants are equal to the third-order moments. Fourth-order cumulants are given in terms of the moments [39] as,…”

Section: Analytical Formulation Of Scalar Mfs For Mildly Non-gaussian...mentioning

confidence: 99%

“…In this paper, we examine in detail the non-Gaussian nature and SI of the Haslam map using the Minkowski tensors as a unified statistical tool. Further, we calculate the generalized skewness and kurtosis cumulants that enter in the perturbative expansion of scalar MFs for weakly non-Gaussian fields about the zeroth-order Gaussian expressions [38,39]. We compare the non-Gaussian deviations of the MFs that are obtained using the analytic expressions with the exact numerical calculations.…”

mentioning

confidence: 99%

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

Rahman,

Chingangbam,

Ghosh

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

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

Rahman

¹

,

Chingangbam

²

,

Ghosh

³

2021

J. Cosmol. Astropart. Phys.

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 5 publications

References 85 publications

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: second-order effects

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: second-order effects

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

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

Contact Info

Product

Resources

About