HierArchical Wavelet Coefficients on the Sphere (HAWCS): the Scattering Transform on Spherical Maps

Extracting Gaussian information from data is well understood, but characterizing non-Gaussianity is challenging. We introduce HAWCS, a healpy-based Python package for efficiently computing wavelet scattering transform coefficients from full-sky maps. These coefficients provide compact summary statistics that are sensitive to higher-order structure and interactions across angular scales. We test the implementation on controlled Gaussian and non-Gaussian fields and demonstrate its computational efficiency. We then apply HAWCS to thermal Sunyaev-Zeldovich maps derived from Planck observations and four simulations, quantifying and comparing their higher-order statistical properties. We also use the method to expose the signatures of gravitational lensing in cosmic microwave background (CMB) temperature maps. These results demonstrate that HAWCS is a practical tool for validating component-separation pipelines and cosmological simulations, and a promising summary statistic for characterizing non-Gaussian structures beyond the power spectrum. These coefficients could also be used to constrain or generate simulated maps that reproduce the same statistical features of real data.