The Maximum Angular-Diameter Distance in Cosmology
  Unlike other observational signatures in cosmology, the angular-diameter
distance d_A(z) uniquely reaches a maximum (at z_max) and then shrinks to zero
towards the big bang. The location of this turning point depends sensitively on
the model, but has been difficult to measure. In this paper, we estimate and
use z_max inferred from quasar cores: (1) by employing a sample of 140 objects
yielding a much reduced dispersion due to pre-constrained limits on their
spectral index and luminosity, (2) by reconstructing d_A(z) using Gaussian
processes, and (3) comparing the predictions of seven different cosmologies and
showing that the measured value of z_max can effectively discriminate between
them. We find that z_max=1.70 +\- 0.20---an important new probe of the
Universe's geometry. The most strongly favoured model is R_h=ct, followed by
Planck LCDM. Several others, including Milne, Einstein-de Sitter and Static
tired light are strongly rejected. According to these results, the R_h=ct
universe, which predicts z_max=1.718, has a ~92.8% probability of being the
correct cosmology. For consistency, we also carry out model selection based on
d_A(z) itself. This test confirms that R_h=ct and Planck LCDM are among the few
models that account for angular-size data better than those that are
disfavoured by z_max. The d_A(z) comparison, however, is less discerning than
that with z_max, due to the additional free parameter, H_0. We find that
H_0=63.4 +\- 1.2 km/s/Mpc for R_h=ct, and 69.9 +\- 1.5 km/s/Mpc for LCDM. Both
are consistent with previously measured values in each model, though they
differ from each other by over 4 sigma. In contrast, model selection based on
z_max is independent of H_0.
