Homology reveals significant anisotropy in the cosmic microwave background

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 $\chi^2$-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.