Measurements of the cosmological evolution of magnetic fields with the Square Kilometre Array

We investigate the potential of the Square Kilometre Array (SKA) for measuring the magnetic fields in clusters of galaxies via Faraday rotation of background polarized sources. The populations of clusters and radio sources are derived from an analytical cosmological model, combined with an extrapolation of current observational constraints. We adopt an empirical model for the Faraday screen in individual clusters, gauged to observations of nearby clusters and extrapolate the polarization properties for the radio source population from the National Radio Astronomy Observatory Very Large Array Sky Survey. We find that about 10 per cent of the sky is covered by a significant extragalactic Faraday screen. Most of it has rotation measures between 10 and 100 rad m-2. We argue that the cluster centres should have up to about 5000 rad m-2. We show that the proposed mid frequency aperture array of the SKA as well as the lowest band of the SKA dish array are well suited to make measurements for most of these rotation measure values, typically requiring a signal-to-noise ratio of 10. We calculate the spacing of sources forming a grid for the purpose of measuring foreground rotation measures: it reaches a spacing of 36 arcsec for a 100 h SKA observation per field. We also calculate the statistics for background rotation measure (RM) measurements in clusters of galaxies. We find that a first phase of the SKA would allow us to take stacking experiments out to high redshifts (> 1), and provide improved magnetic field structure measurements for individual nearby clusters. The full SKA aperture array would be able to make very detailed magnetic field structure measurements of clusters with more than 100 background sources per cluster up to a redshift of 0.5 and more than 1000 background sources per cluster for nearby clusters, and could for reasonable assumptions about future measurements of electron densities in high-redshift clusters constrain the power-law index for the magnetic field evolution to better than delta m = 0.4, if the magnetic field in clusters should follow B proportional to (1 + z)m.