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We present a multi-scale topological analysis of the temperature fluctuationmaps from the NPIPE and FFP10 datasets, invoking relative homology to accountfor the analysis in the presence of masks. For the topological components, wedetect a $2.96\sigma$ deviation between the observations and simulations at $N= 128, FWHM = 80'$, for the FFP10 dataset. For the topological loops, weobserve a high deviation between the observation and simulations in the numberof loops at $FWHM = 320'$, at a low dimensionless threshold $\nu = -2.5$, forthe NPIPE dataset. Under a Gaussian assumption, this would amount to adeviation of $\sim 4\sigma$ . However, the distribution in this bin ismanifestly non-Gaussian and does not obey Poisson statistics either. In theabsence of a true theoretical understanding, we simply note that thesignificance is higher than what may be resolved by $600$ simulations. TheFFP10 dataset, indicates a $2.77\sigma$ deviation at this resolution andthreshold. The Euler characteristic reflects the deviations in the componentsand loops. To assess the significance of combined levels for a given scale, weemployed the empirical and theoretical versions of the $\chi^2$ test as well asthe nonparametric Tukey depth test. Although all statistics exhibit a stabledistribution, we favor the empirical version of the $\chi^2$ test in the finalinterpretation, as it indicates the most conservative differences. Even thoughboth datasets exhibit mild to significant discrepancies, they also exhibitcontrasting behaviors at various instances. Therefore, we do not find itfeasible to convincingly accept or reject the null hypothesis. Disregarding thelarge-scale anomalies that persist at similar scales in WMAP and Planck,observations of the cosmic microwave background are largely consistent with thestandard cosmological model within $2\sigma$.

astronomy and astrophysics

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