Spatially Resolving the Kinematics of the $\lesssim 100\,\mu$as Quasar
  Broad-line Region Using Spectroastrometry II. The First Tentative Detection
  in a Luminous Quasar at $z=2.3$
  Direct measurements of the masses of supermassive black holes (SMBHs) are key
to understanding their growth and constrain their symbiotic relationship to
their host galaxies. However, current methods used to directly measure black
hole masses in active quasars become challenging or impossible beyond
$z\gtrsim0.2$. Spectroastrometry (SA) measures the spatial centroid of an
object's spectrum as a function of wavelength, delivering angular resolution
far better than the point-spread function (PSF) for high signal-to-noise ratio
observations. We observed the luminous quasar SDSS J212329.47--005052.9 at
$z=2.279$ with the aim of resolving its $\sim100\mu\mathrm{as}$ H$\alpha$ broad
emission-line region (BLR), and present the first SA constraints on the size
and kinematic structure of the BLR. Using a novel pipeline to extract the SA
signal and reliable uncertainties, we achieved a centroiding precision of
$\simeq100\mu\mathrm{as}$, or $>2000\times$ smaller than the $K$-band
AO-corrected PSF, yielding a tentative $3.2\sigma$ detection of an SA signal
from the BLR. Modeling the BLR emission as arising from an inclined rotating
disk with a mixture of coherent and random motions we constrain
$r_\mathrm{BLR}=454^{+565}_{-162}\,\mu\mathrm{as}$
($3.71^{+4.65}_{-1.28}\,\mathrm{pc}$), providing a $95\%$ confidence upper
limit on the black hole mass $M_\mathrm{BH}\,\sin^2\,i \leq1.8
\times10^9\,\mathrm{M}_\odot$. Our results agree with the $r_\mathrm{BLR}-L$
relation measured for lower-$z$ quasars, but expands its dynamic range by an
order of magnitude in luminosity. We did not detect the potentially stronger SA
signal from the narrow-line region, but discuss in detail why it may be absent.
Already with existing instrumentation, SA can deliver $\sim6\times$ smaller
uncertainties ($\sim15\,\mu\mathrm{as}$) than achieved here, enabling
$\sim10\%$ measurements of SMBH masses in high-$z$ quasars.
