Magnetic Field Evolution in Giant Radio Relics using the example of CIZA
  J2242.8+5301
  Giant radio relics are the arc-shaped diffuse radio emission regions observed
in the outskirts of some merging galaxy clusters. They are believed to trace
shock-waves in the intra-cluster medium. Recent observations demonstrated that
some prominent radio relics exhibit a steepening above 2 GHz in their radio
spectrum. This challenges standard theoretical models because shock
acceleration is expected to accelerate electrons to very high energies with a
power-law distribution in momentum. In this work we attempt to reconcile these
data with the shock-acceleration scenario. We propose that the spectral
steepening may be caused by the highest energy electrons emitting
preferentially in lower magnetic fields than the bulk of synchrotron bright
electrons in relics. Here, we focus on a model with an increasing mag- netic
field behind the shock front, which quickly saturates and then declines. We
derive the time-evolution of cosmic-ray electron spectra in time variable
magnetic fields and an expanding medium. We then apply the formalism on the
large radio relic in the cluster CIZA J2242.8+5301 (the Sausage relic). We show
that under favourable circumstances of magnetic field amplification downstream,
our model can explain the observed radio spectrum, the brightness profile and
the spectral index profile of the relic. A possible interpretation for the
required amplification of the magnetic field downstream is a dynamo acting
behind the shock with an injection scale of magnetic turbulence of about 10
kpc. Our models require injection efficiencies of CRe - which are in tension
with simple diffusive shock acceleration from the thermal pool. We show that
this problem can likely be alleviated considering pre-existing CRe.
