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 polarised sources. [...] 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/m. We argue that the cluster centres
should have up to about 5000 rad/m/m. 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 of ten. 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 hour SKA observation per
field. We also calculate the statistics for background 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 dm=0.4, if the magnetic field
in clusters should follow B ~ (1+z)^m.
