Fast computation of temperature and polarization coupling matrices

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ID: 316557
2026
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Abstract
Abstract We present a fast and exact method for computing CMB mode–coupling matrices based on an optimised evaluation of Wigner–3j symbols. The method exploits analytic structure in the relevant Wigner–3j symbol configurations appearing in temperature and polarization coupling matrices, expressing all required quantities in terms of a small set of recurrence-generated values which are precomputed and stored in lookup tables. This approach reduces the computational cost of constructing the full coupling matrices whilst maintaining numerical accuracy. We demonstrate the performance of the threej_cosmoimplementation using realistic survey masks from current CMB experiments. Relative to standard recursion-based approaches used in existing pseudo-Cℓ pipelines, the method achieves speedups of 6–25 × in practical coupling-matrix constructions, with the largest gains occurring at high multipoles. The algorithm admits efficient parallelisation on both CPUs and GPUs, the latter providing additional acceleration, up to a further $\mathcal {O}(50)$ on modern hardware, without altering the underlying formalism. Beyond full matrix construction, the approach is naturally suited to applications in which only a restricted set of ℓ3 modes is required for each (ℓ1, ℓ2) pair, such as in the computation of band-limited coupling matrices and analytic covariance terms. These features make threej_cosmoa practical backend for pseudo-Cℓ estimation and related calculations in next-generation CMB analysis pipelines.
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openalex_W7164043722 Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Georgia Kiddier, Steven Gratton
Journal monthly notices of the royal astronomical society
Year 2026
DOI
10.1093/mnras/stag1075
URL
https://doi.org/10.1093/mnras/stag1075
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