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Peer-reviewedCC BY 4.0The Astrophysical Journal

Capturing Statistical Isotropy Violation with Rotational Averages

  • Vaishali R1
  • Dipayan Mukherjee1
  • Akshank Tyagi1
  • Tarun Souradeep1,2
  1. 1Raman Research Institute
  2. 2Inter University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune-411007,
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@article{vaishali2026capturing,
  title = {{Capturing Statistical Isotropy Violation with Rotational Averages}},
  author = {Vaishali R and Dipayan Mukherjee and Akshank Tyagi and Tarun Souradeep},
  journal = {The Astrophysical Journal},
  year = {2026},
  month = nov,
  doi = {10.3847/1538-4357/aeaf53},
  url = {https://iopscience.iop.org/article/10.3847/1538-4357/aeaf53},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Capturing Statistical Isotropy Violation with Rotational Averages
AU  - Vaishali R
AU  - Dipayan Mukherjee
AU  - Akshank Tyagi
AU  - Tarun Souradeep
PY  - 2026
DA  - 2026/11/04
JO  - The Astrophysical Journal
DO  - 10.3847/1538-4357/aeaf53
UR  - https://iopscience.iop.org/article/10.3847/1538-4357/aeaf53
AB  - 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.
SN  - 1538-4357
SN  - 0004-637X
ER  -