Ie. Segal et al., STATISTICALLY EFFICIENT PARALLEL TESTING OF THE HUBBLE AND LUNDMARK LAWS IN THE PREEVOLUTIONARY X-RAY-BAND, The Astrophysical journal, 431(1), 1994, pp. 52-68
The Hubble (linear) and Lundmark (homogeneous quadratic) redshift-dist
ance laws are subjected to statistically efficient parallel tests on t
he basis of the complete sample of X-ray active galactic nuclei of Gio
ia et al. (1990), Maccacaro et al. (1991), and Stocke et al. (1991). T
he samples treated to this end consist of bright flux-limited subsampl
es in a variety of redshift ranges. Luminosity functions are estimated
by an optimal non-parametric procedure that makes no assumptions rega
rding the spatial distribution of the sources and is compatible with m
issing redshifts. It assumes only that the discrimination against obje
cts on the basis of flux is inconsequential above a suitable limit, wh
ich may depend on the position of the object. This is subject to a pos
teriori testing, which is provided. The Hubble and Lundmark laws are t
ested in successively higher redshift ranges up to z = 0.4, within whi
ch range Maccacaro et al. find only weak evolution. Directly observed
quantities are compared with the predictions of both cosmologies, and
objective statistical significance levels estimated. The Hubble law ap
pears inconsistent with the observations, unless evolution effectively
equivalent to its replacement by the Lundmark law is postulated. For
example, the dispersion in log flux invariably exceeds the observed va
lue, as does also the correlation of log luminosity with redshift. The
Lundmark law fits the observations closely and moreover predicts that
the predictions of the Hubble law will be deviant in the amounts obse
rved. Very bright optical subsamples give similar results. In the full
redshift range of the sample, z < 3, the chronometric predictions for
directly observed quantities, which involve no adjustable parameters
such as q0, and which assume no evolution, are quite accurate. The lar
ge deviations of the Friedmann predictions from these values, which ma
y be ascribed to unobservable evolutionary effects, are statistically
coincident with the chronometric prediction for the results of analysi
s predicated on Friedmann cosmology.