# 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 »

- Authors:

- Indian Institute of Astrophysics, Koramangala II Block, Bangalore 560 034 (India)
- 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}

}