Pseudo-$C_\ell$s for spin-$s$ fields with component-wise weighting

TL;DR

This paper generalizes the pseudo-$C_ll$ method for power spectrum estimation to handle spin-$s$ fields with complex, anisotropic weighting, enabling more accurate analysis of polarized and shear data.

Contribution

It introduces a new framework for pseudo-$C_ll$ estimation that accounts for component-wise weights and anisotropic noise in spin-$s$ fields, extending standard methods.

Findings

Derived general expressions for mode-coupling coefficients.

Validated the approach with simulations and implementation in NaMaster.

Achieved computational efficiency comparable to standard methods.

Abstract

We present a generalisation of the standard pseudo-$C_\ell$ approach for power spectrum estimation to the case of spin-$s$ fields weighted by a general positive-definite weight matrix that couples the different spin components of the field (e.g. $Q$ and $U$ maps in CMB polarisation analyses, or $\gamma_1$ and $\gamma_2$ shear components in weak lensing). Relevant use cases are, for example, data with significantly anisotropic noise properties, or situations in which different masks must be applied to the different field components. The weight matrix map is separated into a spin-0 part, which corresponds to the "mask" in the standard pseudo-$C_\ell$ approach, and a spin-$2s$ part sourced solely by the anisotropic elements of the matrix, leading to additional coupling between angular scales and $E/B$ modes. The general expressions for the mode-coupling coefficients involving the power spectra of these anisotropic weight components are derived and validated. The generalised algorithm is as computationally efficient as the standard approach. We implement the method in the public code NaMaster.