Spatial homogeneity and redshift-distance laws.

Spatial homogeneity in the radial direction of low-redshift galaxies is subjected to Kafka-Schmidt V/V(m) tests using well-documented samples. Homogeneity is consistent with the assumption of the Lundmark (quadratic redshift-distance) law, but large deviations from homogeneity are implied by the assumption of the Hubble (linear redshift-distance) law. These deviations are similar to what would be expected on the basis of the Lundmark law. Luminosity functions are obtained for each law by a nonparametric statistically optimal method that removes the observational cutoff bias in complete samples. Although the Hubble law correlation of absolute magnitude with redshift is reduced considerably by elimination of the bias, computer simulations show that its bias-free value is nevertheless at a satistically quite significant level, indicating the self-inconsistency of the law. The corresponding Lundmark law correlations are quite satisfactory satistically. The regression of redshift on magnitude also involves radial spatial homogeneity and, according to R. Soneira, has slope determining the redshift-magnitude exponent independently of the luminosity function. We have, however, rigorously proved the material dependence of the regression on this function and here exemplify our treatment by using the bias-free functions indicated, with results consistent with the foregoing argument.