A self-consistent framework to study magnetic fields with strong gravitational lensing and polarised radio sources

We introduce a unified approach that, given a strong gravitationally lensed polarised source, self-consistently infers its complex surface brightness distribution and the lens galaxy mass-density profile, magnetic field and electron density from interferometric data. The method is fully Bayesian, pixellated and three-dimensional: the source light is reconstructed in each frequency channel on a Delaunay tessellation with a magnification-adaptive resolution. We tested this technique using simulated interferometric observations with a realistic model of the lens, for two different levels of source polarisation and two different lensing configurations. For all data sets, the presence of a Faraday rotating screen in the lens is supported by the data with strong statistical significance. In the region probed by the lensed images, we can recover the Rotation Measure and the parallel component of the magnetic field with an average error between 0.6 and 11 rad m$^{-2}$ and 0.3 and 3 nG, respectively. Given our choice of model, we find the electron density is the least well-constrained component due to a degeneracy with the magnetic field and disk inclination. The background source total intensity, polarisation fraction, and polarisation angle are inferred with an error between 4 and 10 per cent, 15 and 50 per cent, and 1 to 12 degrees, respectively. Our analysis shows that both the lensing configuration and the intrinsic model degeneracies play a role in the quality of the constraints that can be obtained.