The redshift-distance relation.

Key predictions of the Hubble law are inconsistent with direct observations on equitable complete samples of extragalactic sources in the optical, infrared, and x-ray wave bands-e.g., the predicted dispersion in apparent magnitude is persistently greatly in excess of its observed value, precluding an explanation via hypothetical perturbations or irregularities. In contrast, the predictions of the Lundmark (homogeneous quadratic) law are consistent with the observations. The Lundmark law moreover predicts the deviations between Hubble law predictions and observation with statistical consistency, while the Hubble law provides no explanation for the close fit of the Lundmark law. The flux-redshift law F [symbol, see text] (1 + z)/z appears consistent with observations on equitable complete samples in the entire observed redshift range, when due account is taken of flux limits by an optimal statistical method. Under the theoretical assumption that space is a fixed sphere, as in the Einstein universe, this law implies the redshift-distance relation z = tan2(r/2R), where R is the radius of the spherical space. This relation coincides with the prediction of chronometric cosmology, which estimates R as 160 +/- 40 Mpc (1 parsec = 3.09 x 10(16) m) from the proper motion to redshift relation of superluminal sources. Tangential aspects, including statistical methodology, fundamental physical theory, bright cluster galaxy samples, and proposed luminosity evolution, are briefly considered.