# Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background

## TL;DR

This study investigates the topology of temperature fluctuations in the CMB using Betti numbers, revealing deviations from Gaussian simulations and suggesting potential implications for primordial non-Gaussianity or non-trivial topology models.

## Contribution

It introduces the use of Betti numbers and relative homology to analyze CMB topology, highlighting significant deviations from Gaussian LCDM simulations at specific scales.

## Key findings

- Significant deviations between observed and simulated Betti numbers at 2-7 degrees.
- Detection of mildly unusual Euler characteristic behavior at 3.66 degrees.
- Strong differences in non-parametric tests across most scales.

## 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 LCDM paradigm with Gaussian distributed fluctuations. The survey of the CMB over $\mathbb{S}^2$ 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 $\chi^2$-test shows differences between observations and simulations, yielding $p$-values at per-cent to less than per-mil levels roughly between 2 to 7 degrees. The highest observed deviation for $b_0$ and $b_1$ is approximately between $3\sigma$-4$\sigma$ at scales of 3 to 7 degrees. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66 degrees in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor WMAP satellite. The mildly anomalous behaviour of Euler characteristic is related to the strongly anomalous behaviour of components and holes. These are also the scales at which the observed maps exhibit low variance compared to the simulations. Non-parametric tests show even stronger differences at almost all scales. 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 to look at primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology.