Angular Diameter Distances (z ≈ 1 Peak)

The angular diameter distance d_A(z) is the rare cosmological function with a built-in landmark: it rises with redshift, peaks at a turnover z_max, then shrinks toward the Big Bang, because the universe was physically smaller when distant light departed. The location of that maximum is a clean discriminant among cosmologies: flat Planck ΛCDM places it near z_max = 1.59, while alternative expansion histories shift it substantially. Using 140 compact quasar cores as standard rulers and a model-independent Gaussian-process reconstruction, the turnover has been measured at z_max = 1.70 +/- 0.20 (arXiv:1807.07548), and the same analysis finds the data favor alternative expansion histories over Planck ΛCDM, which is not the best-fitting model of those examined.

The discomfort is twofold. First, the measured turnover sits high of the ΛCDM expectation, mildly but in the same direction as the quasar Hubble-diagram deviation: the constant-Lambda expansion history appears to mis-shape distances precisely in the redshift range between the supernova-calibrated low-z regime and the CMB anchor at z = 1100, the long interval where the model is interpolating rather than measured. Second, ΛCDM has no flexibility to respond: once its parameters are fixed by the CMB and BAO, the entire d_A(z) curve including z_max is rigidly determined, so a confirmed turnover offset cannot be absorbed, only denied via systematics in the ruler population.

The standing is early but sharpening: the compact-radio-source ruler carries evolution and selection systematics under active debate, and the 0.20 uncertainty leaves ΛCDM within reach. The method scales, though, and combined Euclid BAO, DESI full-shape, and larger VLBI core samples will localize the turnover to a few percent, turning the d_A peak into a precision shape test of the expansion history in exactly the redshift band where evolving dark energy hints already live.

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