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Title: Tensor Minkowski Functionals: first application to the CMB

Abstract

Tensor Minkowski Functionals (TMFs) are tensor generalizations of the usual Minkowski Functionals which are scalar quantities. We introduce them here for use in cosmological analysis, in particular to analyze the Cosmic Microwave Background (CMB) radiation. They encapsulate information about the shapes of structures and the orientation of distributions of structures. We focus on one of the TMFs, namely W {sub 2}{sup 1,1}, which is the (1,1) rank tensor generalization of the genus. The ratio of the eigenvalues of the average of W {sub 2}{sup 1,1} over all structures, α, encodes the net orientation of the structures; and the average of the ratios of the eigenvalues of W {sub 2}{sup 1,1} for each structure, β, encodes the net intrinsic anisotropy of the structures. We have developed a code that computes W {sub 2}{sup 1,1}, and from it α and β, for a set of structures on the 2-dimensional Euclidean plane. We use it to compute α and β as functions of chosen threshold levels for simulated Gaussian and isotropic CMB temperature and E mode fields. We obtain the value of α to be one for both temperature and E mode, which means that we recover the statistical isotropy of density fluctuationsmore » that we input in the simulations. We find that the standard ΛCDM model predicts a charateristic shape of β for temperature and E mode as a function of the threshold, and the average over thresholds is β∼ 0.62 for temperature and β∼ 0.63 for E mode. Accurate measurements of α and β can be used to test the standard model of cosmology and to search for deviations from it. For this purpose we compute α and β for temperature and E mode data of various data sets from PLANCK mission. We compare the values measured from observed data with those obtained from simulations to which instrument beam and noise characteristics of the 44GHz frequency channel have been added (which are provided as part of the PLANCK data release). We find very good agreement of β and α between all PLANCK temperature data sets with ΛCDM expectations. E mode data show good agreement for β but α for all data sets deviate from ΛCDM predictions higher than 3−σ. It is most likely that the deviations are probing the anisotropy of the noise field and beam characteristics of the detector rather than the true E mode signal since for 44GHz the signal-to-noise ratio is well below one. This will be further investigated after the full PLANCK data becomes publicly available.« less

Authors:
 [1];  [2]
  1. Indian Institute of Astrophysics, Koramangala II Block, Bangalore 560 034 (India)
  2. Indian Institute of Science, C.V. Raman Ave, Bangalore 560 012 (India)
Publication Date:
OSTI Identifier:
22676162
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 06; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ANISOTROPY; COMPARATIVE EVALUATIONS; COSMOLOGY; DENSITY; DISTRIBUTION; EIGENVALUES; EUCLIDEAN SPACE; FLUCTUATIONS; FORECASTING; ISOTROPY; MINKOWSKI SPACE; RELICT RADIATION; SIGNAL-TO-NOISE RATIO; SIMULATION; STANDARD MODEL; TWO-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS

Citation Formats

Ganesan, Vidhya, and Chingangbam, Pravabati, E-mail: vidhya@iiap.res.in, E-mail: prava@iiap.res.in. Tensor Minkowski Functionals: first application to the CMB. United States: N. p., 2017. Web. doi:10.1088/1475-7516/2017/06/023.
Ganesan, Vidhya, & Chingangbam, Pravabati, E-mail: vidhya@iiap.res.in, E-mail: prava@iiap.res.in. Tensor Minkowski Functionals: first application to the CMB. United States. doi:10.1088/1475-7516/2017/06/023.
Ganesan, Vidhya, and Chingangbam, Pravabati, E-mail: vidhya@iiap.res.in, E-mail: prava@iiap.res.in. Thu . "Tensor Minkowski Functionals: first application to the CMB". United States. doi:10.1088/1475-7516/2017/06/023.
@article{osti_22676162,
title = {Tensor Minkowski Functionals: first application to the CMB},
author = {Ganesan, Vidhya and Chingangbam, Pravabati, E-mail: vidhya@iiap.res.in, E-mail: prava@iiap.res.in},
abstractNote = {Tensor Minkowski Functionals (TMFs) are tensor generalizations of the usual Minkowski Functionals which are scalar quantities. We introduce them here for use in cosmological analysis, in particular to analyze the Cosmic Microwave Background (CMB) radiation. They encapsulate information about the shapes of structures and the orientation of distributions of structures. We focus on one of the TMFs, namely W {sub 2}{sup 1,1}, which is the (1,1) rank tensor generalization of the genus. The ratio of the eigenvalues of the average of W {sub 2}{sup 1,1} over all structures, α, encodes the net orientation of the structures; and the average of the ratios of the eigenvalues of W {sub 2}{sup 1,1} for each structure, β, encodes the net intrinsic anisotropy of the structures. We have developed a code that computes W {sub 2}{sup 1,1}, and from it α and β, for a set of structures on the 2-dimensional Euclidean plane. We use it to compute α and β as functions of chosen threshold levels for simulated Gaussian and isotropic CMB temperature and E mode fields. We obtain the value of α to be one for both temperature and E mode, which means that we recover the statistical isotropy of density fluctuations that we input in the simulations. We find that the standard ΛCDM model predicts a charateristic shape of β for temperature and E mode as a function of the threshold, and the average over thresholds is β∼ 0.62 for temperature and β∼ 0.63 for E mode. Accurate measurements of α and β can be used to test the standard model of cosmology and to search for deviations from it. For this purpose we compute α and β for temperature and E mode data of various data sets from PLANCK mission. We compare the values measured from observed data with those obtained from simulations to which instrument beam and noise characteristics of the 44GHz frequency channel have been added (which are provided as part of the PLANCK data release). We find very good agreement of β and α between all PLANCK temperature data sets with ΛCDM expectations. E mode data show good agreement for β but α for all data sets deviate from ΛCDM predictions higher than 3−σ. It is most likely that the deviations are probing the anisotropy of the noise field and beam characteristics of the detector rather than the true E mode signal since for 44GHz the signal-to-noise ratio is well below one. This will be further investigated after the full PLANCK data becomes publicly available.},
doi = {10.1088/1475-7516/2017/06/023},
journal = {Journal of Cosmology and Astroparticle Physics},
number = 06,
volume = 2017,
place = {United States},
year = {Thu Jun 01 00:00:00 EDT 2017},
month = {Thu Jun 01 00:00:00 EDT 2017}
}
  • In this paper, we investigate the utility of Minkowski Functionals as a probe of cold/hot disk-like structures in the CMB. In order to construct an accurate estimator, we resolve a long-standing issue with the use of Minkowski Functionals as probes of the CMB sky — namely that of systematic differences (''residuals'') when numerical and analytical MF are compared. We show that such residuals are in fact by-products of binning, whereas it was originally attributed to pixelation or masking effects. We then derive a map-independent estimator that encodes the effects of binning, applicable to beyond our present work. Using this residual-freemore » estimator, we show that small disk-like effects (as claimed by Vielva et al. [1,2]) can be detected only when a large sample of such maps are averaged over. In other words, our estimator is noise-dominated for small disk sizes at WMAP resolution. To confirm our suspicion, we apply our estimator to the WMAP7 data to obtain a null result.« less
  • The detection of the magnetic type B -mode polarization is the main goal of future cosmic microwave background (CMB) experiments. In the standard model, the B -mode map is a strong non-gaussian field due to the CMB lensing component. Besides the two-point correlation function, the other statistics are also very important to dig the information of the polarization map. In this paper, we employ the Minkowski functionals to study the morphological properties of the lensed B -mode maps. We find that the deviations from Gaussianity are very significant for both full and partial-sky surveys. As an application of the analysis,more » we investigate the morphological imprints of the foreground residuals in the B -mode map. We find that even for very tiny foreground residuals, the effects on the map can be detected by the Minkowski functional analysis. Therefore, it provides a complementary way to investigate the foreground contaminations in the CMB studies.« less
  • The measurement of cosmic shear using weak gravitational lensing is a challenging task that involves a number of complicated procedures. We study in detail the systematic errors in the measurement of weak-lensing Minkowski Functionals (MFs). Specifically, we focus on systematics associated with galaxy shape measurements, photometric redshift errors, and shear calibration correction. We first generate mock weak-lensing catalogs that directly incorporate the actual observational characteristics of the Canada-France-Hawaii Lensing Survey (CFHTLenS). We then perform a Fisher analysis using the large set of mock catalogs for various cosmological models. We find that the statistical error associated with the observational effects degradesmore » the cosmological parameter constraints by a factor of a few. The Subaru Hyper Suprime-Cam (HSC) survey with a sky coverage of ∼1400 deg{sup 2} will constrain the dark energy equation of the state parameter with an error of Δw {sub 0} ∼ 0.25 by the lensing MFs alone, but biases induced by the systematics can be comparable to the 1σ error. We conclude that the lensing MFs are powerful statistics beyond the two-point statistics only if well-calibrated measurement of both the redshifts and the shapes of source galaxies is performed. Finally, we analyze the CFHTLenS data to explore the ability of the MFs to break degeneracies between a few cosmological parameters. Using a combined analysis of the MFs and the shear correlation function, we derive the matter density Ω{sub m0}=0.256±{sub 0.046}{sup 0.054}.« less
  • Analytic formulas of Minkowski functionals in two-dimensional random fields are derived, including effects of second-order non-Gaussianity in the presence of both the bispectrum and trispectrum. The set of formulas provides a promising method to constrain the primordial non-Gaussianity of the universe by temperature fluctuations in the cosmic microwave background radiation. In a case of local-type non-Gaussianity, the Minkowski functionals are analytically given by powers of quadratic and cubic parameters, f{sub NL} and g{sub NL} (and {tau}{sub NL}). Our formulas are not restricted to this particular model, and applicable to a wide class of non-Gaussian models. The analytic formulas are comparedmore » to numerical evaluations from non-Gaussian realizations of temperature maps, showing very good agreement.« less
  • We study the cosmological information contained in the Minkowski functionals (MFs) of weak gravitational lensing convergence maps. We show that the MFs provide strong constraints on the local-type primordial non-Gaussianity parameter f {sub NL}. We run a set of cosmological N-body simulations and perform ray-tracing simulations of weak lensing to generate 100 independent convergence maps of a 25 deg{sup 2} field of view for f {sub NL} = -100, 0 and 100. We perform a Fisher analysis to study the degeneracy among other cosmological parameters such as the dark energy equation of state parameter w and the fluctuation amplitude {sigma}{submore » 8}. We use fully nonlinear covariance matrices evaluated from 1000 ray-tracing simulations. For upcoming wide-field observations such as those from the Subaru Hyper Suprime-Cam survey with a proposed survey area of 1500 deg{sup 2}, the primordial non-Gaussianity can be constrained with a level of f {sub NL} {approx} 80 and w {approx} 0.036 by weak-lensing MFs. If simply scaled by the effective survey area, a 20,000 deg{sup 2} lensing survey using the Large Synoptic Survey Telescope will yield constraints of f {sub NL} {approx} 25 and w {approx} 0.013. We show that these constraints can be further improved by a tomographic method using source galaxies in multiple redshift bins.« less