2 Clustering analysis

2.1 The sample and the diagnostic method

Table 1 shows the redshifts of the EROs identified in the K20 survey (C02) and classified as old passively evolving or dusty-SF galaxies, sorted with increasing redshift and divided between the two survey fields (32.2 arcmin² from CDFS and 19.8 arcmin² from 0055-27). The classification of EROs as old galaxies is based on the detection of the 4000 Å break and CaII H&K absorption with undetected (or very weak) [OII]$\lambda$3727 emission, while objects with strong [OII]$\lambda$3727 emission and an absence of a distinctive 4000 Å break were assigned to the dusty-SF class (see C02 for details).

Despite being by far the largest sample of EROs with identified redshifts, standard methods for evaluating the full two point correlation function cannot be still applied because of the small number of objects. Nevertheless the clustering properties of the old and dusty-SF samples can be investigated by studying the frequency of close pairs. This kind of approach has been applied in regimes with limited amount of information, e.g. to early studies of QSO clustering (Shaver 1984, cf. also Hartwick & Schade 1990), or to analyses of the arrival directions of ultra high energy cosmic rays (Tinyakov & Tkachev 2001), and relates to the integral under the correlation function on small scales, where most of the amplitude lies.

 

 

Table 1: ERO redshifts in the K20 survey. All but four EROs have $K\leq 19.2$. The redshift measurement errors are preliminarily estimated to be of the order of $\sigma \sim 100$-200 kms^-1.

CDFS		0055-27
Old	Dusty-SF	Old	Dusty-SF
0.726	0.796	0.790	0.820
1.0191	0.863	0.864	0.996
1.039	0.891	0.896	1.210
1.096	0.974	0.896	1.240
1.215	0.9961	0.935	1.3001
1.222	1.030	1.050	1.419
1.222	1.094	1.104
	1.1091	1.166
	1.149
	1.221
	1.294
	1.327

$\parbox{6cm}{ \footnotesize{$^1$\space Objects with $19.2<K\leq20$ }.}$

From Table 1, it can be noted that the sample of old EROs contains two pairs that, within the observational redshift accuracy, have the same redshift (z=0.896in the 0055-27 field and z=1.222 in the CDFS), with an additional object close to the second pair at z=1.215. On the other hand, the sample of dusty-SF EROs contains no really close pair, the closest pair having a relatively large redshift separation $\Delta z = 0.015$ (z=1.094 and z=1.109 in the CDFS, corresponding to $\Delta v\sim 4500$ kms^-1). The two old ERO pairs with the same redshift have also quite small angular separations ( $1\hbox{$^\prime$ }$), implying spatial separations of 0.51 and 0.82 h^-1 Mpc, while the two closest dusty-SF pairs are separated by 24 and 40 h^-1 Mpc, respectively. The number of independent pairs in the samples is 81 for the dusty-SF EROs and 49 for the old EROs, thus immediately suggesting a higher intrinsic clustering amplitude for the old EROs.

To assess the significance of observed pair counts we first generate random samples. The selection functions are constructed from the observed redshift distributions of the two ERO populations. Simulated samples were built by assigning at random a redshift (rounded to $\Delta z=0.001$ to match the data redshift measurements) extracted from the appropriate selection function, with sky positions within boundaries matching the area of each of our fields, and number of objects as in the relative observations (Table 1). The resulting probability of finding by chance $\geq$2 pairs of old EROs within a separation $\leq$0.82 h^-1Mpc is about $5\times 10^{-5}$, a clear evidence of clustering among the sample of old EROs. On the other hand, the probabilities of finding the closest dusty-SF ERO pair at $\leq$24 h^-1 Mpc and the two closest pairs at $\leq$40 h^-1 Mpc are both $\sim$$97\%$, consistent with purely random chance.

Figure 1: For each panel we show the redshift distribution of 100 objects extracted from our simulations, that incorporates a given correlation length r0, as indicated on the figure, illustrating the influence of the correlation amplitude on the overall smoothness of pencil beam redshift surveys (see text).

2.2 Comparison to clustered samples

We now proceed a step further and generate simulated samples incorporating a known 2-point clustering amplitude, in order to derive information on the clustering of the two classes, and to obtain meaningful estimates of the variance inherent in the pairs statistics in the sample. We follow the recipe described in D01, based on the Soneira & Peebles (1977, 1978) prescription, allowing us to generate many samples with a given value of r₀ over very large volumes. We adopt the canonical parameterisation $\xi(r)\propto r^{-\gamma}$ with a slope of $\gamma=1.8$ (justified by the observed angular slope $\delta=0.8$, D00) for the 2-point correlation function and allow the amplitude to vary.

For these simulations one has also to account for the redshift space distortion, which tends to decrease the numbers of small scale pairs, and for the measurement error in the redshift. For the pairwise peculiar velocity dispersion we adopt the local value of $\sigma_{12}=360$ kms^-1 (Landy et al. 1998, see also Peacock et al. 2001) and their functional parameterisation, which is assumed not to evolve significantly over the redshift range of our data (e.g. Kauffmann et al. 1999). For the redshift error $\sigma=150$ kms^-1 is adopted (cf. Table 1), and we note that its contribution is small compared to the peculiar velocity term. To each simulated object, an error in the redshift measurement and a peculiar velocity is added in quadrature, chosen randomly from the appropriate distributions, before rounding its redshift to $\Delta z=0.001$ to match the data redshift measurements.

Figure 2: Top panel: the cumulative distribution of pair separations observed for the old EROs (heavy line with filled circles). The horizontal error bars show the $2\sigma$ range estimated from our simulations with random (solid lines) and clustered (dotted lines, r0=10 h-1 Mpc) realizations. Bottom panel: the same but for the dusty-SF EROs. This comparison shows that while the error on the estimate of the correlation length of either sample is quite broad, it is clear that the dusty-SF EROs as a class are completely inconsistent with a correlation length of order 10 h-1 Mpc, estimated from projected samples of EROs (D01, Firth et al. 2001).

As expected, the close pairs statistics is strongly dependent on the correlation length. For example, Fig. 1 shows that in the case of strong clustering almost all the objects reside in spikes with 2 or more objects in each $\Delta z=0.001$ bin, and therefore even with our small number of objects we would expect to find a number of very close pairs (as indeed we do find for the old EROs). In fact, with the clustered samples the probability to find $\geq$2 pairs within $\leq$0.82 h^-1 Mpc increases strongly with r₀ and at the $1\sigma$ level the observed close pairs statistics requires the correlation length of the present sample of old EROs to lie in the broad range h^-1 Mpc . On the other hand, for the dusty-SF EROs, the observation of the two closest pairs within 40 h^-1 Mpc constrains r₀<2.5 h^-1 Mpc at the $3\sigma$ confidence level. Figure 2 summarises concisely the comparison between the fraction of observed pairs below a given scale compared with the random (r₀=0) and clustered (r₀=10 h^-1 Mpc) expectations for a range of scales.

2.3 Spatial and angular clustering of $\mathsfsl{K\leq19.2}$ EROs

If we assume r₀<2.5 h^-1 Mpc for the observed sample of dusty-SF EROs, this results in an angular clustering amplitude $A(1^{\rm o})\lesssim 0.002$at $K\sim19$. We recall that EROs as a whole (including both old and dusty-SF objects) have a factor of 10 larger angular amplitude than this (D00). A solid result of this analysis is therefore that the dusty-SF EROs cannot be the cause of the strong angular clustering of EROs reported by D00, in agreement with the considerations of D01. It is clear from our redshift survey that a significant fraction of EROs are weakly clustered dusty-SF galaxies which therefore dilutes the true angular clustering amplitude of the early-type galaxies population responsible for the majority of the clustering signal. A detailed estimate of the amplitude of this dilution effect would need a more precise knowledge of the relative fractions of both classes and a measure of the cross correlation between the two ERO species. In fact, two dusty-SF EROs are in close redshift pairs with old EROs (see Table 1), with $\Delta z \leq 0.002$ and distances within 3.2 h^-1 Mpc, with a probability of only 2% to happen by chance. This is evidence of some positive cross-correlation bewteen the two ERO species, an intriguing result considering the different physical properties of the two populations. We defer a discussion of this aspect as a part of the ongoing analysis of the clustering of the whole K20 sample (Daddi et al. 2002, in preparation). The cross correlation term will tend to reduce the dilution effect of the dusty-SF EROs to the angular clustering of all EROs. In any case, although for $z\sim1$ early-type galaxies the spatial clustering amplitude of $r_0=12\pm3$ h^-1 Mpc (derived in D01) is more secure, being based on a relatively large sample, and consistent with the present analysis, it is likely that such amplitude should be revised upward in light of the findings presented here.

2.4 Analysis of systematic effects

We tested the stability of these results with respect to the statistical uncertainty in the shapes of the selection functions, which mostly influences the numbers of widely separated pairs. A change in the pairwise peculiar velocity dispersion $\sigma_{12}$ by 20%, would result in a change of only about 10% for the estimated r₀ values, influencing of course the analysis of both ERO species in the same direction and thus leaving the result unchanged. Figure 2 shows that the two closest pairs for the old EROs in our survey are expected at $\lesssim$5 h^-1 Mpc separation at the $2\sigma$ level for $r_0\sim10$ h^-1 Mpc, thus demonstrating that our result would hold correctly even if, because of redshift errors and roundings, the two closest pairs had been found at $\Delta z=0.001$-0.002. The effect of redshift errors is in fact negligible for the dusty-SF ERO pairs, being all of them at >20 h^-1 Mpc separation. Finally, we tested that the result is stable to variations of the color threshold at least up to R-K>4.5.