Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**9.3. Value of H_{0}**

In these pages
Hodge (1981)
reviewed the debate up to ~ 1980 on the
value of the Hubble constant, giving extensive references. Since that
time attempts have been made to discover the source of the factor of 2
difference in *H*_{0} between the long- and the short-
distance scale
workers. A factor of 2 in distance at a given redshift is equivalent
to 1.5 magnitude (factor of 4 in luminosity). The source of the
discrepancy must be sought and convincingly found if the matter is to
be put to rest while members of the present generation are still active.

The factor of 2 is *not* in the local calibrators
(Tammann 1987a,
b),
where at most only ~ 0.3 mag separates the various adopted
calibrations. The explanation is almost certainly to be found as bias
effects in the analysis of data from flux-limited catalogs, no matter
what distance indicator is adopted, if that indicator is anything
other than redshift. The proof of this statement is the *apparent
increase of H_{0} with redshift,* ranging from 50 km
s

There have been many ways to discuss this type of Malmquist bias in
flux-limited samples, most of which are complicated enough to be but
dimly understood except, perhaps, by their authors.
Figure 15
illustrates still another discussion of the effect and its
consequences. Shown is a schematic *m*(*z*) Hubble diagram
using objects
that have a spread in absolute magnitude *M*. Parallel envelope lines
drawn to encompass the sample define the loci of absolute magnitudes
*M*_{1} and *M*_{2}. (For a linear
velocity-distance relation, recall that the
slope of these lines is
*dm*/*d* log *z* = 5.) Divide the data into redshift
zones, labeled 1-10 and shown hatched in Figure 15.
If the sample is
taken from a magnitude-limited catalog, there will be a limit line as
shown, for which no objects in the sample are fainter. Consider now
the mean (ridge) line of the data sample in the *z*, *m*
plane. The true mean <*M*_{T}> value is seen in the
data up to distance interval 4-5, but
beyond that the data begin to be systematically too bright compared
with the *true* value of <*M*_{T}>.

What are the consequences? Suppose we assign an absolute magnitude
<*M*_{T}> to each galaxy in the sample. For those
closer than distance
interval 5-6 we will make as many positive as negative errors in the
correct *m* - *M* distance modulus due to the luminosity
spread in *M*. This
part of the sample is distance limited. However, the flux limitation
of the fainter sample progressively removes fainter absolute
magnitudes from the remaining set as the true distance increases,
giving false (biased) mean *m* - *M* values. The inferred
distances *are too small* in the mean for this subsample (because the
<*M*_{T}> used is too
faint to apply to it, as shown by the position of the crosses). By
using these incorrect inferred photometric distances to obtain
*H*_{0},
progressively larger (incorrect) values of *H*_{0} would be
obtained as an
artifact. Note that the correct value of *H*_{0} is that
obtained only in the distance interval closer than 5-6.

Detailed analysis of actual data
(Sandage 1988a,
b)
has shown this
bias explicitly in samples of ScI galaxies and in galaxies used for
the Tully-Fisher distance scale method. By reading the data in the
redshift limit of
*z* 0,
low values of *H*_{0} have been obtained using the
local calibrators with Cepheid distances. This is equivalent to the
method of
Richter & Huchtmeier
(1984),
who restrict their sample to the distance-limited list of
Kraan-Korteweg &
Tammann (1979)
and obtain a low value of *H*_{0} directly (cf.
Sandage 1988b).

From these studies, together with a cluster data analysis with the Tully-Fisher method (Kraan-Korteweg et al. 1988), the value of the Hubble constant has been found to be in the range

where the formal error is adopted to be ~ ± 10% from individual studies. The value found from supernovae of type Ia, using both an empirical (Sandage & Tammann 1982) and a theoretical calibration (Sutherland & Wheeler 1984, Arnett et al. 1985) of the absolute magnitude, gives (Cadonau et al. 1985)

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which in turn gives a Hubble time of
*H*_{0}^{-1} = 22.7 ± 5.5 Gyr.