Nature of Faint Radio Sources in GOODS-North and GOODS-South Fields - I.
  Spectral Index and Radio-FIR Correlation
  We present the first results from the deep and wide 5 GHz radio observations
of the Great Observatories Origins Deep Survey (GOODS)-North ($\sigma=3.5 \;
\mu Jy \; beam^{-1}$, synthesized beam size $\theta =$ 1.47 arcsec $\times$
1.42 arcsec, and 52 sources over 109 arcmin$^{2}$) and GOODS-South ($\sigma=3.0
\; \mu Jy \; beam^{-1}$, $\theta=$0.98 arcsec $\times$ 0.45 arcsec, and 88
sources over 190 arcmin$^{2}$) fields using the Karl G. Jansky Very Large
Array. We derive radio spectral indices {\alpha} between 1.4 and 5 GHz using
the beam-matched images and show that the overall spectral index distribution
is broad even when the measured noise and flux bias are considered. We also
find a clustering of faint radio sources around $\alpha=0.8$, but only within
$S_{5GHz} < 150 \; \mu Jy$. We demonstrate that the correct radio spectral
index is important for deriving accurate rest frame radio power and analyzing
the radio-FIR correlation, and adopting a single value of $\alpha=0.8$ leads to
a significant scatter and a strong bias in the analysis of the radio-FIR
correlation, resulting from the broad and asymmetric spectral index
distribution. When characterized by specific star formation rates, the
starburst population (58%) dominates the 5 GHz radio source population, and the
quiescent galaxy population (30%) follows a distinct trend in spectral index
distribution and the radio-FIR correlation. Lastly, we offer suggestions on
sensitivity and angular resolution for future ultra-deep surveys designed to
trace the cosmic history of star formation and AGN activity using radio
continuum as a probe.
