Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the $\texttt{NPIPE}$ and $\texttt{FFP10}$ data releases

We present a multi-scale topological analysis of the temperature fluctuation maps from the NPIPE and FFP10 datasets, invoking relative homology to account for the analysis in the presence of masks. For the topological components, we detect a $2.96\sigma$ deviation between the observations and simulations at $N = 128, FWHM = 80'$, for the FFP10 dataset. For the topological loops, we observe a high deviation between the observation and simulations in the number of loops at $FWHM = 320'$, at a low dimensionless threshold $\nu = -2.5$, for the NPIPE dataset. Under a Gaussian assumption, this would amount to a deviation of $\sim 4\sigma$ . However, the distribution in this bin is manifestly non-Gaussian and does not obey Poisson statistics either. In the absence of a true theoretical understanding, we simply note that the significance is higher than what may be resolved by $600$ simulations. The FFP10 dataset, indicates a $2.77\sigma$ deviation at this resolution and threshold. The Euler characteristic reflects the deviations in the components and loops. To assess the significance of combined levels for a given scale, we employed the empirical and theoretical versions of the $\chi^2$ test as well as the nonparametric Tukey depth test. Although all statistics exhibit a stable distribution, we favor the empirical version of the $\chi^2$ test in the final interpretation, as it indicates the most conservative differences. Even though both datasets exhibit mild to significant discrepancies, they also exhibit contrasting behaviors at various instances. Therefore, we do not find it feasible to convincingly accept or reject the null hypothesis. Disregarding the large-scale anomalies that persist at similar scales in WMAP and Planck, observations of the cosmic microwave background are largely consistent with the standard cosmological model within $2\sigma$.