# Unexpected topology of the temperature fluctuations in the cosmic microwave background

## Abstract

We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the *Planck* satellite with a thousand simulated maps generated according to the $Λ$CDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over S ^{2}is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric *χ* ^{2}-test shows differences between observations and simulations, yielding *p*-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3 *σ* sporadically at these scales. The highest observed deviation between the observations and simulations for *b* _{0}and *b* _{1}is approximately between 3 *σ* and 4 *σ* at scales of 3–7°. There aremore »

- Authors:

- Univ. Lyon (France); Israel Inst. of Technology, 32000 Haifa (Israel)
- Israel Inst. of Technology, 32000 Haifa (Israel)
- Univ. Lyon (France)
- Inst. of Science and Technology Austria
- Kapteyn Astronomical Inst. (The Netherlands)
- Univ. of California, San Diego, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1577583

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Astronomy and Astrophysics

- Additional Journal Information:
- Journal Volume: 627; Journal ID: ISSN 0004-6361

- Publisher:
- EDP Sciences

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS; Astronomy & Astrophysics; cosmic background radiation; early Universe; methods: statistical; methods: numerical

### Citation Formats

```
Pranav, Pratyush, Adler, Robert J., Buchert, Thomas, Edelsbrunner, Herbert, Jones, Bernard J. T., Schwartzman, Armin, Wagner, Hubert, and van de Weygaert, Rien. Unexpected topology of the temperature fluctuations in the cosmic microwave background. United States: N. p., 2019.
Web. doi:10.1051/0004-6361/201834916.
```

```
Pranav, Pratyush, Adler, Robert J., Buchert, Thomas, Edelsbrunner, Herbert, Jones, Bernard J. T., Schwartzman, Armin, Wagner, Hubert, & van de Weygaert, Rien. Unexpected topology of the temperature fluctuations in the cosmic microwave background. United States. doi:10.1051/0004-6361/201834916.
```

```
Pranav, Pratyush, Adler, Robert J., Buchert, Thomas, Edelsbrunner, Herbert, Jones, Bernard J. T., Schwartzman, Armin, Wagner, Hubert, and van de Weygaert, Rien. Wed .
"Unexpected topology of the temperature fluctuations in the cosmic microwave background". United States. doi:10.1051/0004-6361/201834916. https://www.osti.gov/servlets/purl/1577583.
```

```
@article{osti_1577583,
```

title = {Unexpected topology of the temperature fluctuations in the cosmic microwave background},

author = {Pranav, Pratyush and Adler, Robert J. and Buchert, Thomas and Edelsbrunner, Herbert and Jones, Bernard J. T. and Schwartzman, Armin and Wagner, Hubert and van de Weygaert, Rien},

abstractNote = {We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the $Λ$CDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over S2is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yieldingp-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations forb0andb1is approximately between 3σ and 4σ at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck’s predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.},

doi = {10.1051/0004-6361/201834916},

journal = {Astronomy and Astrophysics},

number = ,

volume = 627,

place = {United States},

year = {2019},

month = {7}

}

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