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Astron. Astrophys. 360, 861-870 (2000)

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4. The reality of the galaxy sets

In this section we assign likelihoods to the 19 sets of galaxies which have been identified in the previous section. If [FORMULA] is the incompleteness of our redshift sample in the magnitude interval [FORMULA] (see Fig. 8), then the redshift distribution reads

[EQUATION]

where [FORMULA] is the volume element at the redshift z and [FORMULA] is the luminosity function (LF hereafter). We take the LF in the [FORMULA] band given by Postman et al. (1996) and convert it to our [FORMULA] band, using the Postman et al. transformation procedure. Our reference LF has therefore a Schechter (1976) form with parameters [FORMULA] and [FORMULA]. We also assume negligible evolution of the LF out to [FORMULA] (see Lilly et al. 1995and Lin et al. 1999). Our LF then depends on z only through the evolution of the stellar populations and the K-correction, that we take from Poggianti (1997).

[FIGURE] Fig. 8. The function [FORMULA], computed as the ratio between the binned magnitude distributions of galaxies with redshifts, and of galaxies with magnitudes, in the range [FORMULA], in the six EIS cluster fields. The dashed line is the best fit to the data with a Schechter function.

In Fig. 9, we plot [FORMULA], computed according to Eq. (1), along with the observed z-distribution. Based on the estimated [FORMULA], we find that 16 out of 19 sets correspond to significant overdensities in redshift space, with a probability [FORMULA]%. Since Eq. (1) provides [FORMULA] for a uniform galaxy distribution, these probabilities do not include the effect of large scale clustering (see, e.g., Zaritsky et al. 1997). In order to account for the large scale clustering, and following the approach by Holden et al. (1999a), we modulate [FORMULA] with the redshift distribution derived from the CFRS assuming our selection function, [FORMULA].

[FIGURE] Fig. 9. The redshift distribution function, [FORMULA] (dashed line) and the redshift histogram of the 67 galaxies of our sample in the redshift range [FORMULA]. [FORMULA] is normalised to the same number of objects, 67.

We first convert the CFRS [FORMULA] magnitudes to the [FORMULA] system (see Lilly et al. 1995). We then extract 50000 galaxies from the CFRS, with a bootstrap sampling, adopting the magnitude distribution of our sample (see Fig. 7). A Kolmogorov-Smirnov test shows that the bootstrapped CFRS and our data-set have similar redshift distributions. This is expected since magnitude selection is the main process leading to the inclusion of galaxies in the sample.

We extract random subsamples of 11 galaxies from the bootstrapped CFRS reference sample (11 is the mean number of galaxies with redshift in our EIS cluster fields). Using the same procedure described in Sect. 3, we identify sets of galaxies within these subsamples, and compute their probabilities relative to the uniform redshift distribution [FORMULA] given by Eq. (1). In this way we construct a distribution of probabilities to detect a real system within a galaxy sample which includes large-scale clustering, but not galaxy clusters. We finally obtain the likelihoods of the 19 observed sets, by comparing their original [FORMULA]-based probabilities to the distribution of probabilities for the random sets. These 19 likelihoods are listed in Table 3.

We find that four of our 19 sets have a likelihood [FORMULA]%; all of them have at least three galaxy members. The four sets are flagged in the last column of Table 3. We refer to these four sets as the `real systems' hereafter. As expected, many of the sets with a significant overdensity with respect to the uniform redshift distribution are no longer significant when compared to a redshift distribution which includes the large scale clustering.

Our results are robust against modifications of the adopted LF (we change [FORMULA] by [FORMULA], and [FORMULA] by [FORMULA] mag), and of the galaxy-type for which the evolutionary- and K-corrections are computed. Furthermore, we verify that varying cosmological parameters (h75, [FORMULA] and [FORMULA]) within conservative ranges, only induces marginal changes in the system likelihoods. Finally, we also checked that narrowing the [FORMULA] range used to define the redshift-sets, from 0.015 to 0.010, hardly modifies the membership and likelihoods of the sets.

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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