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 ΛCDM 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 𝕊2 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 χ2-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 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is 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.
Highlights
The Λ cold dark matter standard paradigm of cosmology postulates that the Universe consists primarily of cold non-relativistic dark matter, which reveals its presence only through gravitational interactions, and the Universe is currently driven by dark energy, causing accelerated volume expansion in this model
The simulations are based on the ΛCDM paradigm and assume that the temperature fluctuations have a Gaussian distribution
We compute the absolute homology for the dataset, and confirm that the results are consistent with those from relative homology
Summary
The Λ cold dark matter (or ΛCDM) standard paradigm of cosmology postulates that the Universe consists primarily of cold non-relativistic dark matter, which reveals its presence only through gravitational interactions, and the Universe is currently driven by dark energy, causing accelerated volume expansion in this model. The ΛCDM paradigm together with the inflationary theories in their simplest forms, predict the primordial perturbations to be realizations of a homogeneous and isotropic Gaussian random field (Guth & Pi 1982) This hypothesis is supported experimentally by CMB observations (Smoot et al 1992; Bennett et al 2003; Spergel et al 2007; Komatsu et al 2011; Planck Collaboration XIII 2016) and theoretically by the central limit theorem. It is related to but different from geometry, which measures size and shape Both geometry and topology have been used in the past to study the structure of the CMB radiation and other cosmic fields. To count them in the presence of regions with unreliable data, we compute the ranks of the homology groups relative to the mask that covers these regions
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