Analytic detection thresholds for measurements of linearly polarized intensity using rotation measure synthesis

A fully analytic statistical formalism does not yet exist to describe radio wavelength measurements of linearly polarized intensity that are produced using rotation measure synthesis. In this work we extend the analytic formalism for standard linear polarization, namely that describing measurements of the quadrature sum of Stokes Q and U intensities, to the rotation measure synthesis environment. We derive the probability density function and expectation value for Faraday-space polarization measurements for both the case where true underlying polarized emission is present within unresolved Faraday components, and for the limiting case where no such emission is present. We then derive relationships to quantify the statistical significance of linear polarization measurements in terms of standard Gaussian statistics. The formalism developed in this work will be useful for setting signal-to-noise ratio detection thresholds for measurements of linear polarization, for the analysis of polarized sources potentially exhibiting multiple Faraday components and for the development of polarization debiasing schemes.