Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps

Schmalzing, Jens; Górski, K. M.

doi:10.1046/j.1365-8711.1998.01467.x

Cited by 232 publications

(251 citation statements)

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“…In this paper these derivatives are only used to determine the skeleton and to locate the stationary points, but, as shown by Schmalzing & Górski (1998), it is possible to express all the Minkowski functionals in terms of the field itself and its firstand second-order derivatives. It is most convenient not to compute the covariant derivatives of the temperature anisotropy field directly from the pixel values, but rather to do it from the expansion of the anisotropy field in spherical harmonics,…”

Section: Locatinggthe Stationary Points Of a Randommentioning

confidence: 99%

“…The next step is to expand the excluded regions of the mask in all directions as a safeguard against border effects. This is done according to the procedures of Schmalzing & Górski (1998); the mask map consisting of zeros and ones is convolved with a Gaussian beam of the required FWHM, and the new, expanded mask is then determined by including all pixels with values higher than some given threshold, for instance, 0.95 or 0.99. For maximum safety we perform this operation twice, each time with a limiting threshold of 0.99.…”

Section: The Wmap Data and The Simulationsmentioning

confidence: 99%

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Testing for Non‐Gaussianity in theWilkinson Microwave Anisotropy ProbeData: Minkowski Functionals and the Length of the Skeleton

Eriksen

Novikov

Lilje

et al. 2004

ApJ

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174

216

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The three Minkowski functionals and the recently defined length of the skeleton are estimated for the co-added first-year Wilkinson Microwave Anisotropy Probe (WMAP) data and compared with 5000 Monte Carlo simulations, based on Gaussian fluctuations with the a priori best-fit running-index power spectrum and WMAP-like beam and noise properties. Several power spectrum-dependent quantities, such as the number of stationary points, the total length of the skeleton, and a spectral parameter, , are also estimated. While the area and length Minkowski functionals and the length of the skeleton show no evidence for departures from the Gaussian hypothesis, the northern hemisphere genus has a 2 that is large at the 95% level for all scales. For the particular smoothing scale of 3N40 FWHM it is larger than that found in 99.5% of the simulations. In addition, the WMAP genus for negative thresholds in the northern hemisphere has an amplitude that is larger than in the simulations with a significance of more than 3 . On the smallest angular scales considered, the number of extrema in the WMAP data is high at the 3 level. However, this can probably be attributed to the effect of point sources. Finally, the spectral parameter is high at the 99% level in the northern Galactic hemisphere, while perfectly acceptable in the southern hemisphere. The results provide strong evidence for the presence of both nonGaussian behavior and an unexpected power asymmetry between the northern and southern hemispheres in the WMAP data. Subject headingg s: cosmic microwave background -cosmology: observations -methods: statistical

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Section: Locatinggthe Stationary Points Of a Randommentioning

confidence: 99%

Section: The Wmap Data and The Simulationsmentioning

confidence: 99%

Testing for Non‐Gaussianity in theWilkinson Microwave Anisotropy ProbeData: Minkowski Functionals and the Length of the Skeleton

Eriksen

Novikov

Lilje

et al. 2004

ApJ

Self Cite

174

216

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“…The area, length, and level-crossing statistics directly quantify the amount of contour surfaces (Ryden 1988b;Ryden et al 1989;Torres 1994). The Minkowski functionals (Mecke, Buchert, & Wagner 1994;Kerscher et al 1998;Bharadwaj et al 2000), which are closely related to the above statistics, are also applied to smoothed cosmic fields (Winitzki & Kosowsky 1997;Naselsky & Novikov 1998;Schmalzing & Gorski 1998;Schmalzing et al 1999;Schmalzing & Diaferio 2000). These statistics of smoothed density fields are considered as powerful descriptors of the statistical information of the universe.…”

Section: Introductionmentioning

confidence: 99%

Statistics of Smoothed Cosmic Fields in Perturbation Theory. I. Formulation and Useful Formulae in Second‐Order Perturbation Theory

Matsubara¹

2003

ApJ

131

166

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We formulate a general method for perturbative evaluations of statistics of smoothed cosmic fields and provide useful formulae for application of the perturbation theory to various statistics. This formalism is an extensive generalization of the method used by Matsubara, who derived a weakly nonlinear formula of the genus statistic in a three-dimensional density field. After describing the general method, we apply the formalism to a series of statistics, including genus statistics, level-crossing statistics, Minkowski functionals, and a density extrema statistic, regardless of the dimensions in which each statistic is defined. The relation between the Minkowski functionals and other geometrical statistics is clarified. These statistics can be applied to several cosmic fields, including three-dimensional density field, three-dimensional velocity field, twodimensional projected density field, and so forth. The results are detailed for second-order theory of the formalism. The effect of the bias is discussed. The statistics of smoothed cosmic fields as functions of rescaled threshold by volume fraction are discussed in the framework of second-order perturbation theory. In CDMlike models, their functional deviations from linear predictions plotted against the rescaled threshold are generally much smaller than that plotted against the direct threshold. There is still a slight meatball shift against rescaled threshold, which is characterized by asymmetry in depths of troughs in the genus curve. A theory-motivated asymmetry factor in the genus curve is proposed.

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“…Many studies have aimed at characterising the statistical properties of the COBE/DMR data (Amendola 1996;Kogut et al 1996;Ferreira et al 1998;Heavens 1998;Pando et al 1998b;Schmalzing & Gorski 1998;Bromley & Tegmark 1999;Magueijo 2000;Mukherjee et al 2000). These investigations have relied on various statistical indicators both in the real space (e.g.…”

Section: Introductionmentioning

confidence: 99%

Using the COBE/DMR data as a test-bed for normality assessments

Aghanim¹,

Forni²,

Bouchet³

2001

A&A

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Abstract.A very important property of a statistical distribution is to know whether it obeys Gaussian statistics or not. On the one hand, it is of paramount importance in the context of CMB anisotropy studies, since deviations from a Gaussian distribution could indicate the presence of uncorrected measurement systematics, of remaining fluctuations contributed by foregrounds, of deviations from the simplest models of inflation or of topological defects. On the other hand, looking for a non-Gaussian signal is a very ill-defined task and performances of various assessment methods may differ widely when applied in different contexts. In previous studies, we introduced a sensitive wavelet-based method which we apply here to the COBE/DMR data set, already extensively studied, using different approaches. This provides an objective way to compare methods. It turns out that our multi-scale wavelet decomposition is a sensitive method. Yet, we show that the results are rather sensitive to the choice of both the decomposition scheme and the wavelet basis. We find that the detection of the non-Gaussian signature is "marginal" with a probability of at most 99%.

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Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps

Cited by 232 publications

References 32 publications

Testing for Non‐Gaussianity in theWilkinson Microwave Anisotropy ProbeData: Minkowski Functionals and the Length of the Skeleton

Testing for Non‐Gaussianity in theWilkinson Microwave Anisotropy ProbeData: Minkowski Functionals and the Length of the Skeleton

Statistics of Smoothed Cosmic Fields in Perturbation Theory. I. Formulation and Useful Formulae in Second‐Order Perturbation Theory

Using the COBE/DMR data as a test-bed for normality assessments

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