Marginal distributions for cosmic variance limited CMB polarization data

TL;DR

This paper derives computationally convenient expressions for the marginal distributions of the CMB polarization power spectrum, enabling improved sampling and estimation techniques for cosmic variance limited data.

Contribution

It introduces new formulas for marginal distributions of CMB polarization spectra and applies them to develop a novel sampling algorithm and estimators for power spectrum analysis.

Findings

New conditional CMB power spectrum sampling algorithm with flexible binning.

Blackwell-Rao estimators for polarization distributions that converge quickly.

Applicable to a wide range of cosmic variance limited CMB data analyses.

Abstract

We provide computationally convenient expressions for all marginal distributions of the polarization CMB power spectrum distribution P(C_l|sigma_l), where C_l = {C_l^TT, C_l^TE, C_l^EE, C_l^BB} denotes the set of ensemble averaged polarization CMB power spectra, and sigma_l = {sigma_l^TT, sigma_l^TE, sigma_l^EE, sigma_l^BB} the set of the realization specific polarization CMB power spectra. This distribution describes the CMB power spectrum posterior for cosmic variance limited data. The expressions derived here are general, and may be useful in a wide range of applications. Two specific applications are described in this paper. First, we employ the derived distributions within the CMB Gibbs sampling framework, and demonstrate a new conditional CMB power spectrum sampling algorithm that allows for different binning schemes for each power spectrum. This is useful because most CMB experiments have very different signal-to-noise ratios for temperature and polarization. Second, we provide new Blackwell-Rao estimators for each of the marginal polarization distributions, which are relevant to power spectrum and likelihood estimation. Because these estimators represent marginals, they are not affected by the exponential behaviour of the corresponding joint expression, but converge quickly.