Winds as the origin of radio emission in $z=2.5$ radio-quiet extremely
  red quasars
  Most active galactic nuclei (AGNs) are radio-quiet, and the origin of their
radio emission is not well-understood. One hypothesis is that this radio
emission is a by-product of quasar-driven winds. In this paper, we present the
radio properties of 108 extremely red quasars (ERQs) at $z=2-4$. ERQs are among
the most luminous quasars ($L_{bol} \sim 10^{47-48}$ erg/s) in the Universe,
with signatures of extreme ($\gg 1000$ km/s) outflows in their
[OIII]$\lambda$5007 \AA\ emission, making them the best subjects to seek the
connection between radio and outflow activity. All ERQs but one are unresolved
in the radio on $\sim 10$ kpc scales, and the median radio luminosity of ERQs
is $\nu L_\nu [{\rm 6\,GHz}] = 10^{41.0}$ erg/s, in the radio-quiet regime, but
one to two orders of magnitude higher than that of other quasar samples. The
radio spectra are steep, with a mean spectral index $\langle \alpha \rangle =
-1.0$. In addition, ERQs neatly follow the extrapolation of the low-redshift
correlation between radio luminosity and the velocity dispersion of
[OIII]-emitting ionized gas. Uncollimated winds, with a power of one per cent
of the bolometric luminosity, can account for all these observations. Such
winds would interact with and shock the gas around the quasar and in the host
galaxy, resulting in acceleration of relativistic particles and the consequent
synchrotron emission observed in the radio. Our observations support the
picture in which ERQs are signposts of extremely powerful episodes of quasar
feedback, and quasar-driven winds as a contributor of the radio emission in the
intermediate regime of radio luminosity $\nu L_\nu = 10^{39}-10^{42}$ erg/s.
