  Galactic magnetic fields in the local Universe are strong and omnipresent.
There is mounting evidence that galaxies were magnetized already in the early
Universe. Theoretical scenarios including the turbulent small-scale dynamo
predict magnetic energy densities comparable to the one of turbulence. Based on
the assumption of this energy equipartition, we determine the galactic
synchrotron flux as a function of redshift z. The conditions in the early
Universe are different from the present day, in particular the galaxies have
more intense star formation. To cover a large range of conditions we consider
models based on two different systems: one model galaxy comparable to the Milky
Way and one typical high-z starburst galaxy. We include a model of the steady
state cosmic ray spectrum and find that synchrotron emission can be detected up
to cosmological redshifts with current and future radio telescopes. Turbulent
dynamo theory is in agreement with the origin of the observed correlation
between the far-infrared (FIR) luminosity L_FIR and the radio luminosity
L_radio. Our model reproduces this correlation well at z=0. We extrapolate the
FIR-radio correlation to higher redshift and predict a time evolution with a
significant deviation from its present-day appearance already at z~2 for a gas
density that increases strongly with z. In particular, we predict a decrease of
the radio luminosity with redshift which is caused by the increase of cosmic
ray energy losses at high z. The result is an increase of the ratio between
L_FIR and L_radio. Simultaneously, we predict that the slope of the FIR-radio
correlation becomes shallower with redshift. This behavior of the correlation
could be observed in the near future with ultra-deep radio surveys.
