The Radio - X-ray relation as a star formation indicator: Results from
  the VLA--E-CDFS Survey
  In order to trace the instantaneous star formation rate at high redshift, and
hence help understanding the relation between the different emission mechanisms
related to star formation, we combine the recent 4 Ms Chandra X-ray data and
the deep VLA radio data in the Extended Chandra Deep Field South region. We
find 268 sources detected both in the X-ray and radio band. The availability of
redshifts for $\sim 95$ of the sources in our sample allows us to derive
reliable luminosity estimates and the intrinsic properties from X-ray analysis
for the majority of the objects. With the aim of selecting sources powered by
star formation in both bands, we adopt classification criteria based on X-ray
and radio data, exploiting the X-ray spectral features and time variability,
taking advantage of observations scattered across more than ten years. We
identify 43 objects consistent with being powered by star formation. We also
add another 111 and 70 star forming candidates detected only in the radio or
X-ray band, respectively. We find a clear linear correlation between radio and
X-ray luminosity in star forming galaxies over three orders of magnitude and up
to $z \sim 1.5$. We also measure a significant scatter of the order of 0.4 dex,
higher than that observed at low redshift, implying an intrinsic scatter
component. The correlation is consistent with that measured locally, and no
evolution with redshift is observed. Using a locally calibrated relation
between the SFR and the radio luminosity, we investigate the L_X(2-10keV)-SFR
relation at high redshift. The comparison of the star formation rate measured
in our sample with some theoretical models for the Milky Way and M31, two
typical spiral galaxies, indicates that, with current data, we can trace
typical spirals only at z<0.2, and strong starburst galaxies with
star-formation rates as high as $\sim 100 M_\odot yr^{-1}$, up to $z\sim 1.5$.
