Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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4.4. The Cluster Incompleteness Bias

When one constructs the log H-vs-p diagram for members of a galaxy cluster using the direct TF relation, one generally observes log H increasing towards smaller values of the TF parameter p, i.e. towards the approaching magnitude limit. In one distant cluster, with a small number of measured galaxies, this trend may be difficult to recognize, and it is especially hard to see the expected unbiased plateau at large values of p, which gives the correct distance (and log H). Using data from several clusters and normalizing, instead of distance, the parameter p to take into account the vicinity of the magnitude limit, the combined log H-vs-pnorm diagram reveals better the bias behavior. This was shown by Bottinelli et al (1987) using 11 clusters (B-magnitude TF relation), and furthermore by Bottinelli et al (1988b) for clusters with infrared data. In both cases, the distance scale and the Hubble constant were derived in good agreement with the method of normalized distances for field galaxies. Kraan-Korteweg et al (1988), Fouqué et al (1990) analyzed the distance of the Virgo cluster and clearly also gathered evidence of the influence of the cluster incompleteness bias.

A graphic way of determining the distance to a cluster (in fact, solving Kapteyn's formula of Problem I for an ensemble of clusters and deriving the value of Ho) was used in (Bottinelli et al 1988b). Willick 1994 has described an iterative approach.

Recently, Sandage et al (1995) gave a very illuminating exposition of the cluster incompleteness bias, using field galaxies in redshift bins to imitate clusters at different distances. Again, using the principle that adding a fainter sample shows the presence of bias, if it exists, from their Spaenhauer approach, these authors show the presence and behavior of the bias in the imitated clusters. An important empirical result concerns how deep in magnitudes, beyond the brightest galaxy, one must penetrate a cluster so as to avoid the bias. This depends on the dispersion sigma of the TF relation and on the value of the TF parameter p. For small values of p (slowest rotators, faintest galaxies), the required magnitude range may be as wide as 8 mag. It is clear that for distant clusters such deep samples are beyond reach, and one must always be cautious of the incompleteness bias because it always leads to too large a value of Ho if uncorrected.

Another important point emphasized by Sandage et al (1995) is the artificial decrease in the apparent scatter of the TF relation, if derived from magnitude-limited cluster samples. This was also pointed out by Willick (1994), and it was implicit in the conclusion by Bottinelli et al (1988a) that this bias changes the slope of the TF relation. Sandage et al (1995) calculate the dependence of the apparent sigma on the magnitude range Deltam reached for a cluster. For instance, if the true dispersion is 0.6 mag, penetration by Deltam = 3 mag gives sigma approx 0.4 mag, and Deltam = 6 mag is needed to reveal the true sigma. They conclude that the sometimes-mentioned small dispersions for the TF relations below sigma approx 0.4 mag are probably caused in this way (for other viewpoints, see Willick et al 1996, Bernstein et al 1994).

The problem of artificially decreased scatter is a dangerous one because it may lead to the conclusion that the selection bias, which is generally dependent on sigma2, is insignificant. In this manner, the bias itself produces an argument against its presence.

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