Cosmological Analysis Using Spherical Harmonics and Wavelets
Summary
Cosmological analysis through spherical harmonics and wavelets constitutes a powerful framework for probing the large-scale structure and primordial fluctuations of the Universe. Spherical harmonics decompose fields defined on the celestial sphere into modes labelled by angular frequency, revealing statistical anisotropies and power spectra of the cosmic microwave background (CMB). However, purely harmonic approaches lack spatial localisation, limiting the detection of localised features or deviations from Gaussianity. Wavelets on the sphere overcome this by providing joint scale–position analysis, enabling the isolation of compact anomalies, non-Gaussian signatures and directional structures. Modern implementations extend to directional wavelets and needlets, which capture orientation information and enjoy quasi-exponential localisation and asymptotic uncorrelation. These tools have been further enhanced by differentiable transforms, multi-resolution schemes and high-performance computing, facilitating gradient-based inference and full-sky analyses at high angular resolution. Applications range from testing statistical isotropy and constraining cosmological parameters to mapping lensing potentials, polarisation patterns and searching for subtle imprints of primordial physics. The synergy of harmonic and wavelet methods thus provides a comprehensive, multi-scale approach to extract cosmological information with both spectral precision and spatial sensitivity.
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Cosmological Analysis Using Spherical Harmonics and Wavelets publication trend
The graph below shows the total number of articles in cosmological analysis using spherical harmonics and wavelets across all publications each year (not limited to Nature Index journals).
Technical terms
Spherical harmonic: A function on the sphere that serves as an angular basis for decomposing fields into multipole components.
Multipole moment: A coefficient in the spherical harmonic expansion quantifying the contribution of a given angular scale to a field.
Wavelet on the sphere: A localized, scale-dependent function used to analyse spherical data in both spatial and frequency domains.
Needlet: A form of spherical wavelet with quasi-exponential localisation and uncorrelated coefficients, enabling precise multi-scale analysis.
Wigner transform: A generalised Fourier transform on the rotation group SO(3) linking spin fields and spherical harmonics for analysis of polarisation and directional data.
References
- Differentiable and accelerated spherical harmonic and Wigner transforms. Journal of Computational Physics (2024).
- Localisation of directional scale-discretised wavelets on the sphere. Applied and Computational Harmonic Analysis (2018).
- Direction sensitive analysis of higher order jump discontinuities along circles on the sphere. GEM - International Journal on Geomathematics (2023).
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