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 $\Omega_{k}=0.43^{+0.27}_{-0.27}$.