Direct comparison of observed magnitude-redshift relations in complete galaxy samples with systematic predictions of alternative redshift-distance laws

The directly observed average apparent magnitude (or in one case, angular diameter) as a function of redshift in each of a number of large complete galaxy samples is compared with the predictions of hypothetical redshift-distance power laws, as a systematic statistical question. Due account is taken of observational flux limits by an entirely objective and reproducible optimal statistical procedure, and no assumptions are made regarding the distribution of the galaxies in space. The laws considered are of the form z is proportional to r^p^, where r denotes the distance, for p = 1, 2 and 3. The comparative fits of the various redshift-distance laws are similar in all the samples. Overall, the cubic law fits better than the linear law, but each shows substantial systematic deviations from observation. The quadratic law fits extremely well except at high redshifts in some of the samples, where no power law fits closely and the correlation of apparent magnitude with redshift is small or negative. In all cases, the luminosity function required for theoretical prediction was estimated from the sample by the non-parametric procedure ROBUST, whose intrinsic neutrality as programmed was checked by comprehensive computer simulations.