Bayesian Inference of Primordial Magnetic Field Parameters from CMB with Spherical Graph Neural Networks

Deep learning has emerged as a transformative methodology in modern cosmology, providing powerful tools to extract meaningful physical information from complex astronomical data. This paper implements a novel Bayesian graph deep learning framework for estimating key cosmological parameters in a primordial magnetic field (PMF) cosmology from simulated Cosmic Microwave Background (CMB) maps. Our methodology utilizes DeepSphere, a spherical convolutional neural network architecture specifically designed to respect the spherical geometry of CMB data through HEALPix pixelization. To advance beyond deterministic point estimates and enable robust uncertainty quantification, we integrate Bayesian Neural Networks (BNNs) into the framework, capturing aleatoric and epistemic uncertainties that reflect the model confidence in its predictions. The proposed approach demonstrates exceptional performance, achieving R2 scores exceeding 89% for the magnetic parameter estimation. We further obtain well-calibrated uncertainty estimates through post hoc training techniques including Variance Scaling and GPNormal. This integrated DeepSphere-BNNs framework delivers accurate parameter estimation from CMB maps with PMF contributions while providing reliable uncertainty quantification, enabling robust cosmological inference in the era of precision cosmology.