Scale-Dependent Geometric Structure in the CMB: Exploratory Topological Statistics

We perform an exploratory statistical analysis of the Planck SMICA cosmic microwave background temperature map using geometric and topological descriptors including fractal scaling estimators, Minkowski functionals, and Betti curves of excursion sets. Gaussian reference distributions are generated using HEALPix synfast simulations based on the Planck best-fit ΛCDM power spectrum. Across the statistics considered we find results broadly consistent with Gaussian ΛCDM expectations. Topology-based descriptors show mild deviations at the ~2–3σ level in some configurations, but these remain below the threshold required to claim a statistically significant anomaly. This work is intended as an exploratory investigation of multiscale geometric statistics in cosmological datasets.