Polarization measurement analysis

I. Impact of the full covariance matrix on polarization fraction and angle measurements^⋆

¹ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
e-mail: ludovic.montier@irap.omp.eu
² CNRS, IRAP, 9 Av. Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
³ Laboratoire de l’Accélérateur Linéaire, Université Paris-Sud 11, CNRS/IN2P3, Orsay, France
⁴ LERMA/LRA – ENS Paris et Observatoire de Paris, 24 rue Lhormond, 75231 Paris Cedex 05, France

Received: 12 July 2013
Accepted: 18 November 2014

Abstract

With the forthcoming release of high precision polarization measurements, such as from the Planck satellite, the metrology of polarization needs to be improved. In particular, it is important to have full knowledge of the noise properties when estimating polarization fraction and polarization angle, which suffer from well-known biases. While strong simplifying assumptions have usually been made in polarization analysis, we present a method for including the full covariance matrix of the Stokes parameters in estimates of the distributions of the polarization fraction and angle. We thereby quantified the impact of the noise properties on the biases in the observational quantities and derived analytical expressions for the probability density functions of these quantities that take the full complexity of the covariance matrix into account, including the Stokes I intensity components. We performed Monte Carlo simulations to explore the impact of the noise properties on the statistical variance and bias of the polarization fraction and angle. We show that for low variations (< 10%) of the effective ellipticity between the Q and U components around the symmetrical case the covariance matrix may be simplified as is usually done, with a negligible impact on the bias. For S/Ns with intensity lower than 10, the uncertainty on the total intensity is shown to drastically increase the uncertainty of the polarization fraction but not the relative bias of the polarization fraction, while a 10% correlation between the intensity and the polarized components does not significantly affect the bias of the polarization fraction. We compare estimates of the uncertainties that affect polarization measurements, addressing limitations of the estimates of the S/N, and we show how to build conservative confidence intervals for polarization fraction and angle simultaneously. This study, which is the first in a set of papers dedicated to analysing polarization measurements, focuses on the basic polarization fraction and angle measurements. It covers the noise regime where the complexity of the covariance matrix may be largely neglected in order to perform further analysis. A companion paper focuses on the best estimators of the polarization fraction and angle and on their associated uncertainties.