CONFIGURACIÓN DEL MÓDULO PTQ-DNET

It was realized a long time ago (Geller & Beers 1982) that substructure in the galaxy number density distribution on the sky can be more easily identified and evaluated by using contour plots. Geller & Beers constructed very simple maps using an overlap- ping box smoothing technique. At that time, no better methodology was available, and the authors were aware of the limits due to the use of a fixed window width. The main conclusion of this work was the finding of a significant amount of sub- structure in about 40 per cent of the cluster sample. Although these results were heavily criticised by many later studies, it is interesting to note that the fraction of clusters in which the detected substructure is claimed to be ‘significant’ has remained approximately the same. However, different studies have yielded different results for

the same clusters. As we have mentioned in Sec. 5.1, this can originate from the fact that different statistical tests are sensitive to different kinds of substructures (bimodality, groups aligned along the line of sight).

In the last years, improved methods for obtaining maps of projected galaxy posi- tions have been developed. In this analysis we will use theadaptive kernel technique. In a nutshell, the kernel density estimator is a generalisation of the histogram. To build a histogram, one simply places a box over each data point and then sums the boxes. The adaptive kernel method uses of a ‘bump’ whose shape is determined by the kernel function, and whose width is specified by the smoothing parameter. The exact value of the smoothing parameter is determined from the distribution of the neighbouring points.

It is a well–known result that using a fixed smoothing length produces a density estimate that is over–smoothed in high density regions and under–smoothed in low– density regions, because of Poisson noise. It is obvious that an over-smoothing in the high–density region can potentially hide statistically significant substructures. In the adaptive kernel method, this problem is avoided by adjusting the ‘local band–width factors’ (i.e. the smoothing length) across the field to match the local density of events5.

Different implementations of adaptive kernel methods using different kernel func- tions are possible. The technique we use in this analysis has been developed by Pisani (1993, 1996, see also references therein) and employs a Gaussian kernel.

It is possible to obtain more efficient algorithms by choosing other kernel functions (see discussion in Pisani (1996)). However, it has been shown that the choice of the kernel is not critical, as different choices differ only slightly in the form of the contours produced. The algorithm developed by Pisani has the advantage of being non–parametric, which means that the amount of smoothing is dependent solely upon the data points, rather than any prior estimate of what the cluster should look like. In order to avoid excessive under–smoothing that can arise because of Poisson noise, the iterative procedure that determines the optimal kernel size corresponding to each point, is stopped the first time the kernel size falls below a value corresponding to about 100 kpc.

As an example of the performance of this technique, we show in Fig. 5.14 the density map obtained for the cluster cl1216–1201.

It is important to realize that, although adaptive–kernel maps represent a pow- erful visual aid in the detection of substructures in data, the presence of multiple density peaks does not provide any ‘quantitative’ indication about their statistical significance. In order to isolate ‘real’ substructures, we now cross-correlate the sub- structure found in the adaptive kernel maps with the radial velocity information.

As a first step, we assign all the spectroscopically confirmed cluster members (recall 5

5.7 Cluster structure

Figure 5.14: Adaptive kernel map (see text for details) for cl1216–1201. The map

is built considering all the objects retained as probable cluster members on the basis of their photometric redshifts and brighter than 25 in the I–band.

that we define these to be all the galaxies within the redshift interval _±0.015 from the BCG) to the closest peak in the density distribution. In practice, we move each galaxy along the density gradients towards a local density maximum.

Note that because of the noise in the density maps, it may happen that the above procedure finds multiple small peaks closely–spaced. We visually investigate each map and sometimes merge clumps that wejudge not to be statistically significantly different. Note, however, that this occurs in a very small number of cases.

For each peak identified using the above procedure, we compute the local mean recessional velocity and velocity dispersion. In order to assess the statistical signifi- cance of each subclump, we use a bootstrap procedure: we randomly draw, from the actual radial velocity distribution, a number of galaxies equal to the number of galax- ies that are assigned to the subclump under investigation. We repeat the procedure 5000 times for each subclump, and define a subclump to be ‘statistically significant’ if the measured velocity dispersion is at least 1σ less than the most probable value measured using the bootstrap procedure. Only subclumps associated with at least

Cluster name zcl Nclump Ng z¯ σ £ km s−1¤ cl1232–1250∗ 0.5419 1 10 0.5478 510 cl1040–1155 0.7044 1 7 0.7046 152 cl1054–1146 0.6972 1 10 0.6962 261 cl1054–1245 0.7498 0 cl1216–1201 0.7941 1 6 0.8000 437 2 18 0.7873 500

Table 5.3: Number of subclumps detected for each of the clusters used in this

analysis (zcl is the spectroscopic redshift of the cluster). For each of the subclumps

we give the number of galaxies with measured velocity dispersion associated to the subclump, its median redshift and its velocity dispersion. (*) This subclump is not detected by our algorithm, as the associated galaxies are not spatially clustered with respect to the overall distribution of cluster galaxies.

four galaxies with measured velocity dispersion are considered. For each subclump, we also apply a clipping procedure, iteratively rejecting the galaxy with the largest deviation from the median local recessional velocity. In this case, we also update the bootstrap procedure by comparing the mean recessional velocity and velocity disper- sion measured for the subclump under consideration with the values obtained for an equal number of galaxies drawn at random from the actual velocity distribution, after clipping an equal number of objects that deviate most from the median recessional velocity.

Figs. from 5.15 to 5.19 show the results obtained for the five clusters used in this analysis. In each figure, we show the density map obtained using the kernel method estimator; red crosses mark the positions of the local density peaks, while empty circles mark the positions of the galaxies with measured radial velocities. The size of the symbols is proportional to eδ, where δ is the parameter that quantifies the local deviation from the cluster dynamical properties (see Sec. 5.6.2). Solid black lines indicate to which peak each galaxy with measured radial velocity is associated, and filled coloured circles mark the position of galaxies that belong to a statistically significant subclump, as defined above.

The figures show that only for the clusters cl1054–1245 and cl1232–1250 we do find no evidence for any statistically significant subclump. In the other three cases, we find at least one subclump that appears to be statistically significant. The results are summarised in Table 5.3 where we give, for each cluster, its spectroscopic redshift (zcl), the number of subclumps that are found to be statistically significant (Nclump), the number of objects in each of these subclumps (Ng), their mean redshift (¯z), and their estimated velocity dispersion (σ). As an example of the performance of our procedure, we also show in Fig. 5.20 the comparison between the mean redshift and

5.7 Cluster structure

Figure 5.15: Adaptive kernel map (see text for details) for cl1040–1155. The map

is built considering all the objects retained as probable cluster members on the basis of their photometric redshifts and brighter than 25 in the I–band. Red crosses mark the position of the maxima in the density field, and empty circles the position of all the galaxies with measured radial velocity. The size of the symbols is proportional to eδ

. Solid black lines indicate to which peak each galaxy with measured radial velocity is associated, and filled coloured circles mark the position of galaxies that belong to a statistically significant subclump.

velocity dispersion for the statistically significant subclump identified in the cluster cl1040–1155, and the distribution of values found by randomly drawing the same number of objects in this subclump from the actual radial velocity distribution of cluster members. In this particular case, the mean redshift of the subclump does not differ significantly from the redshift of the cluster (i.e. there is no significant offset in velocity), but the velocity dispersion is more than 1σ smaller than the typical value found for an equal number of objects randomly drawn from the actual velocity distribution of cluster members. Therefore, there is a high probability that the object under investigation represents a ‘real’ substructure.

Note that the radial velocity histogram of the cluster cl1232–1250 exhibits an obvi- ous subclump with average recessional velocity of 1.65_×105km s−1. TheP statistics of the Dressler & Schectman test also indicate a complex dynamical structure. How- ever, our algorithm is unable to detect the presence of any significant subclump. This

Figure 5.16: As in Fig. 5.15 but for the cluster cl1054–1146.

5.7 Cluster structure

Figure 5.18: As in Fig. 5.15 but for the cluster cl1216–1202.

Figure 5.20: An illustration of our bootstrap procedure to asses the significance of a subclump identified in the adaptive kernel map. We plot the distributions of the median redshift (left panel) and of the velocity dispersion (right panel) measured for each of the Monte Carlo realizations (see text for details).

is because the galaxies populating the high velocity peak in the velocity histogram are not spatially clustered, as indicated by the red filled circles in Fig. 5.19 that mark the positions of these galaxies. Using our bootstrap procedure on these galaxies, we detect a significant subclump of 10 objects with a mean redshift of 0.5478 and a velocity dispersion of 510 km s−1. This case shows that our algorithm is unable to detect the presence of groups projected along the line–of–sight, if the galaxies in these groups are not spatially segregated on the sky with respect to the overall distribution of cluster galaxies.

In addition, it should be noted that the number and the spatial distribution of galaxies for which we have spectroscopic information, is somewhat limited by the spectroscopic magnitude limit (this roughly corresponds to I= 22 for the high redshift clusters and I= 23 for the low redshift clusters) and by geometrical constraints due to the mask design (see also Sec. 5.9). It could be the case that by increasing the number of objects with spectroscopic information, other significant groups would be detected. The number of subclumps detected for each cluster has then to be taken as a lower limit to the true amount of substructure.