Capturing Statistical Isotropy Violation with Rotational Averages

Recent high-precision cosmological observations have revealed several anomalies in the cosmic microwave background (CMB), indicating possible violations of statistical isotropy (SI)—rotational invariance of statistical measures of a random skymap. The bipolar spherical harmonic (BipoSH) formalism, a widely used complete representation of both the SI and statistical anisotropy (nSI) components of a random skymap, is defined and implemented in the spherical harmonic space. In this work we probe the rotational symmetry violation of the correlation function using weighted rotational averages, providing a geometric interpretation of BipoSH as a real-space operation. We establish that the BipoSH measures are related to averages of the correlation function over all rotated configurations weighted by Wigner matrices. As a demonstration, we consider a phenomenological dipole modulation model and numerically implement the rotational average measures and show their agreement with their harmonic space counterparts. The real-space approach could be advantageous in formulating BipoSH like nSI measures on a partial sky. We envisage broader applications of this framework to the studies of primordial non-Gaussianity, CMB polarisation, weak gravitational lensing and the characterization of general random fields on a sphere.