  A key for understanding the evolution of galaxies and in particular their
star formation history will be future ultra-deep radio surveys. While star
formation rates (SFRs) are regularly estimated with phenomenological formulas
based on the local FIR-radio correlation, we present here a physically
motivated model to relate star formation with radio fluxes. Such a relation
holds only in frequency ranges where the flux is dominated by synchrotron
emission, as this radiation originates from cosmic rays produced in supernova
remnants, therefore reflecting recent star formation. At low frequencies
synchrotron emission can be absorbed by the free-free mechanism. This
suppression becomes stronger with increasing number density of the gas, more
precisely of the free electrons. We estimate the critical observing frequency
below which radio emission is not tracing the SFR, and use the three
well-studied local galaxies M 51, M 82, and Arp 220 as test cases for our
model. If the observed galaxy is at high redshift, this critical frequency
moves along with other spectral features to lower values in the observing
frame. In the absence of systematic evolutionary effects, one would therefore
expect that the method can be applied at lower observing frequencies for high
redshift observations. However, in case of a strong increase of the typical gas
column densities towards high redshift, the increasing free-free absorption may
erase the star formation signatures at low frequencies. At high radio
frequencies both, free-free emission and the thermal bump, can dominate the
spectrum, also limiting the applicability of this method.
