Exploring the radio spectral energy distribution of the ultraluminous
  radio-quiet quasar SDSS J0100+2802 at redshift 6.3
  We report deep Karl G. Jansky Very Large Array (VLA) observations of the
optically ultraluminous and radio-quiet quasar SDSS J010013.02 + 280225.8
(hereafter J0100+2802) at redshift $z=$6.3. We detected the radio continuum
emission at 1.5 GHz, 6 GHz, and 10 GHz. This leads to a radio power-law
spectral index of $\alpha = -0.52\pm0.18$ ($S \propto \nu^{\alpha}$). The radio
source is unresolved in all VLA bands with an upper limit to the size of
$0.2^{\prime \prime}$ (i.e., $\sim$ 1.1 kpc) at 10 GHz. We find variability in
the flux density (increase by $\sim 33\%$) and the spectral index (steepened)
between observations in 2016 and 2017. We also find that the VLA 1.5 GHz flux
density observed in the same year is 1.5 times that detected with the Very Long
Baseline Array (VLBA) in 2016 at the same frequency. This difference suggests
that half of the radio emission from J0100+2802 comes from a compact core
within 40 pc, and the rest comes from the surrounding few kpc area which is
diffuse and resolved out in the VLBA observations. The diffuse emission is four
times brighter than that would be expected if driven by star formation. We
conclude that the central active galactic nucleus is the dominant power engine
of the radio emission in J0100+2802.
