Homology reveals significant anisotropy in the cosmic microwave background

TL;DR

This study tests the statistical isotropy of the CMB within ΛCDM by applying relative and persistent homology to Planck PR3/PR4 temperature maps, focusing on the Betti numbers $b_0$ and $b_1$ of excursion sets on ${ andomS}^2$. It employs a two-pronged normalization strategy (local per-hemisphere vs global full-sky) and analyzes hemispheres and quadrants across multiple smoothing scales, using a detailed TopoS2 pipeline with triangulated sphere representations and upper-star filtrations to obtain persistence diagrams. The authors find hemisphere- and quadrant-specific anomalies, notably a degree-scale north-hemisphere excess of isolated components under local normalization and strong tail discrepancies in the south under global normalization, with the first quadrant contributing most to the discrepancies; χ^2-based p-values corroborate nonrandom deviations. These results point to a potential breakdown of statistical isotropy with important implications for cosmological parameter estimation and the Hubble tension, while leaving open questions about cosmological, foreground, or instrumental origins and motivating future persistence-diagram–level analyses.

Abstract

We test the tenet of statistical isotropy of the standard cosmological model via a homology analysis of the cosmic microwave background temperature maps. Examining small sectors of the normalized maps, we find that the results exhibit a dependence on whether we compute the mean and variance locally from the masked patch, or from the full masked sky. Assigning local mean and variance for normalization, we find the maximum discrepancy between the data and model in the galactic northern hemisphere at more than $3.5$ s.d. for the PR4 dataset at degree-scale. For the PR3 dataset, the C-R and SMICA maps exhibit higher significance than the PR4 dataset at $\sim 4$ and $4.1$ s.d. respectively, however the NILC and SEVEM maps exhibit lower significance at $\sim 3.4$ s.d. The southern hemisphere exhibits high degree of consistency between the data and the model for both the PR4 and PR3 datasets. Assigning the mean and variance of the full masked sky decreases the significance for the northern hemisphere, the tails in particular. However the tails in the southern hemisphere are strongly discrepant at more than $4$ standard deviations at approximately $5$ degrees. The $p$-values obtained from the $χ^2$-statistic exhibit commensurate significance in both the experiments. Examining the quadrants of the sphere, we find the first quadrant to be the major source of the discrepancy. Prima-facie, the results indicate a breakdown of statistical isotropy in the CMB maps, however more work is needed to ascertain the source of the anomaly. Regardless, these map characteristics may have serious consequences for downstream computations such as parameter estimation, and the related Hubble tension.

Figures (14)

• Figure 1: Illustration of objects with different topologies, distinguished by the holes of different dimensions present in them. A disk (panel (a)) is characterized as a single connected object, while a ring (panel (b)) is characterized as a single connected object that forms the boundary of a hole. A sphere (panel (c)) is different from both a disk and a ring, as it is a single connected surface bounding the 3D cavity in the interior, while a torus (panel (d)) is characterized by a surface in the shape of a hollow tube that surrounds a visible hole, as well as the hole in the interior of the tube body, which also doubles up as a cavity or void.
• Figure 2: Excursion sets of the 2-sphere at various thresholds of the temperature function defined on it. For high positive thresholds, the excursion set is predominantly composed of isolated objects or components. For low negative thresholds, the excursion set is composed of a single connected component, with additional punctures or loops. For low enough thresholds, the excursion set completes to form the full $2$-sphere, composed of a single connected component and a void in the interior.
• Figure 3: Illustration of the computational aspects of homology and persistent homology.
• Figure 4: Visualization of the CMB temperature fluctuation field in the northern hemisphere in two different views. The masked area covers the whole southern hemisphere and the relevant parts of the northern hemisphere, dictated by the PR3 temperature common mask. The visualization is based on the PR3 observed map, cleaned using the SMICA component separation pipeline, degraded at $N{\hbox{$N$}} = 128$ and smoothed with a Gaussian kernel of $FWHM = 80'$.
• Figure 5: Graphs of ${b}_{0}{\hbox{${b}_{0}$}}$ and ${b}_{1}{\hbox{${b}_{1}$}}$ for the temperature maps for the NPIPE dataset for the northern (top two rows) and southern (bottom two rows) hemispheres. The mean and variance were computed for each hemisphere locally from the unmasked pixels in that hemisphere. The graphs present the normalized differences, and each panel presents the graphs for a range of degradation and smoothing scales. The mask used is the PR3 temperature common mask.