Pitfalls of statistics-limited X-ray polarization analysis

Abstract

Context. One of the difficulties with performing polarization analysis is that the mean polarization fraction of sub-divided data sets is larger than the polarization fraction for the integrated measurement. The resulting bias is one of the properties of the generating distribution discussed in this work. The limitations of Gaussian approximations in standard analysis based on Stokes parameters for estimating polarization parameters and their uncertainties are explored by comparing with a Bayesian analysis. The effect of uncertainty on the modulation factor is also shown, since it can have a large impact on the performance of gamma-ray burst polarimeters. Results are related to the minimum detectable polarization (MDP), a common figure of merit, making them easily applicable to any X-ray polarimeter. Aims. The aim of this work is to quantify the systematic errors induced on polarization parameters and their uncertainties when using Gaussian approximations and to show when such effects are non-negligible. Methods. The probability density function is used to deduce the properties of reconstructed polarization parameters. The reconstructed polarization parameters are used as sufficient statistics for finding a simple form of the likelihood. Bayes theorem is used to derive the posterior and to include nuisance parameters. Results. The systematic errors originating from Gaussian approximations as a function of instrument sensitivity are quantified here. Different signal-to-background scenarios are considered making the analysis relevant for a large variety of observations. Additionally, the change of posterior shape and instrument performance MDP due to uncertainties on the polarimeteric response of the instrument is shown.