Exploring the Hubble Tension and Spatial Curvature from the Ages of Old
  Astrophysical Objects
  We use the age measurements of 114 old astrophysical objects (OAO) in the
redshift range $0\lesssim z\lesssim 8$ to explore the Hubble tension. The age
of the Universe at any $z$ is inversely proportional to the Hubble constant,
$H_0$, so requiring the Universe to be older than the OAO it contains at any
$z$ will lead to an upper limit on $H_0$. Assuming flat $\Lambda$CDM and
setting a Gaussian prior on the matter density parameter $\Omega_{\rm
m}=0.315\pm0.007$ informed by {\it Planck}, we obtain a 95\% confidence-level
upper limit of $H_0<70.6 \rm{~km} \rm{~s}^{-1} \rm{~Mpc}^{-1}$, representing a
$2\sigma$ tension with the measurement using the local distance ladder. We
find, however, that the inferred upper limit on $H_{0}$ depends quite
sensitively on the prior for $\Omega_{\rm m}$, and the Hubble tension between
early-time and local measurements of $H_{0}$ may be due in part to the
inference of both $\Omega_{\rm m}$ and $H_0$ in {\it Planck}, while the local
measurement uses only $H_{0}$. The age-redshift data may also be used for
cosmological model comparisons. We find that the $R_{\rm h}=ct$ universe
accounts well for the data, with a reasonable upper limit on $H_{0}$, while
Einstein-de Sitter fails to pass the cosmic-age test. Finally, we present a
model-independent estimate of the spatial curvature using the ages of 61
galaxies and the luminosity distances of 1,048 Pantheon Type Ia supernovae.
This analysis suggests that the geometry of the Universe is marginally
consistent with spatial flatness at a confidence level of $1.6\sigma$,
characterized as $\Omega_{k}=0.43^{+0.27}_{-0.27}$.
