  (Abrigded) Observations of galaxies up to z 2 show a tight correlation
between far-infrared and radio continuum emission. We explain the far-infrared
- radio continuum correlation by relating star formation and magnetic field
strength in terms of turbulent magnetic field amplification, where turbulence
is injected by supernova explosions from massive stars. We calculate the
expected amount of turbulence in galaxies based on their star formation rates,
and infer the expected magnetic field strength due to turbulent dynamo
amplification. We estimate the timescales for cosmic ray energy losses via
synchrotron emission, inverse Compton scattering, ionization and bremsstrahlung
emission, probing up to which redshift strong synchrotron emission can be
maintained. We find that the correlation between star formation rate and
magnetic field strength in the local Universe can be understood as a result of
turbulent magnetic field amplification. If the typical gas density in the
interstellar medium increases at high z, we expect an increase of the magnetic
field strength and the radio emission, as indicated by current observations.
Such an increase would imply a modification of the far-infrared - radio
correlation. We expect a breakdown when inverse Compton losses start dominating
over synchrotron emission. For a given star formation surface density, we
calculate the redshift where the breakdown occurs, yielding z (Sigma_SFR/0.0045
M_solar kpc^{-2} yr^{-1})^{1/(6-alpha/2)}. In this relation, the parameter
\alpha describes the evolution of the characteristic ISM density in galaxies as
(1+z)^\alpha. Both the possible raise of the radio emission at high redshift
and the final breakdown of the far-infrared -- radio correlation at a critical
redshift will be probed by the Square Kilometre Array (SKA) and its
pathfinders, while the typical ISM density in galaxies will be probed with
ALMA.
