- 2.1 The sample and the diagnostic method
- 2.2 Comparison to clustered samples
- 2.3 Spatial and angular clustering of EROs
- 2.4 Analysis of systematic effects

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 arcmin^{2} from CDFS and 19.8 arcmin^{2} 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]3727 emission, while objects with strong
[OII]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.

CDFS | 0055-27 | ||

Old |
Dusty-SF |
Old |
Dusty-SF |

0.726 | 0.796 | 0.790 | 0.820 |

1.019^{1} |
0.863 | 0.864 | 0.996 |

1.039 | 0.891 | 0.896 | 1.210 |

1.096 | 0.974 | 0.896 | 1.240 |

1.215 | 0.996^{1} |
0.935 | 1.300^{1} |

1.222 | 1.030 | 1.050 | 1.419 |

1.222 | 1.094 | 1.104 | |

1.109^{1} |
1.166 | ||

1.149 | |||

1.221 | |||

1.294 | |||

1.327 |

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
(*z*=1.094 and *z*=1.109 in the CDFS,
corresponding to
kms^{-1}).
The two old ERO pairs with the same redshift have also quite small
angular separations (
), 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
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 2 pairs of old EROs within a separation 0.82 *h*^{-1}Mpc is about
,
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 24 *h*^{-1} Mpc and the
two closest pairs at 40 *h*^{-1} Mpc are both ,
consistent with purely random chance.

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 *r*_{0} over very large volumes. We adopt
the canonical parameterisation
with a
slope of
(justified by the
observed angular slope
,
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
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
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
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
range estimated from our
simulations with random (solid lines) and clustered (dotted lines,
r_{0}=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
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 2 pairs within 0.82 *h*^{-1} Mpc increases strongly with *r*_{0} and at the
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 *r*_{0}<2.5 *h*^{-1} Mpc at the
confidence level.
Figure 2 summarises concisely the comparison between the
fraction of observed pairs below a given scale compared with
the random (*r*_{0}=0) and clustered (*r*_{0}=10 *h*^{-1} Mpc) expectations for
a range of scales.

If we assume *r*_{0}<2.5 *h*^{-1} Mpc for the observed sample of dusty-SF
EROs, this results in an angular clustering amplitude
at .
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
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
early-type galaxies the
spatial clustering amplitude of
*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.

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
by 20%, would result
in a change of only about 10% for the estimated *r*_{0} 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 5 *h*^{-1}
Mpc separation at the
level for
*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
-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.

Copyright ESO 2002