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.
