A&A 405, 53-72 (2003)
DOI: 10.1051/0004-6361:20030527
R. A. Overzier - H. J. A. Röttgering - R. B. Rengelink - R. J. Wilman
Sterrewacht Leiden, PO Box 9513, 2300 RA, Leiden, The Netherlands
Received 16 August 2002 / Accepted 7 April 2003
Abstract
We have measured the angular correlation function, ,
of radio sources in the 1.4 GHz NVSS and FIRST radio surveys.
Below
the signal is dominated by the size distribution of classical double radio galaxies,
an effect underestimated in some previous studies.
We model the physical size distribution of FRII radio galaxies to account for
this excess signal in .
The amplitude of the true cosmological clustering of radio sources is
roughly constant at
for flux limits of 3-40 mJy,
but has increased to
at 200 mJy.
This can be explained if powerful (FRII) radio galaxies probe significantly more massive structures compared
to radio galaxies of average power at .
This is consistent with
powerful high-redshift radio galaxies generally having massive (forming) elliptical hosts in rich (proto-)cluster environments.
For FRIIs we derive a spatial (comoving) correlation length of
h-1 Mpc. This is
remarkably close to that measured for extremely red objects (EROs) associated with a population of old
elliptical galaxies at
by Daddi et al. (2001). Based on their similar clustering properties, we propose that EROs and
powerful radio galaxies may be the same systems seen at different evolutionary stages.
Their r0 is
higher than that of QSOs at a similar redshift, and comparable to that of bright ellipticals locally. This suggests
that r0 (comoving) of these galaxies has changed little from
to z=0, in agreement with current CDM hierarchical merging
models for the clustering evolution of massive early-type galaxies. Alternatively, the clustering of radio galaxies can be explained by the galaxy
conservation model. This then implies that radio galaxies of average power are the progenitors of the local field population of early-types, while
the most powerful radio galaxies will evolve into a present-day population with r0 comparable to that of local rich clusters.
Key words: cosmology: large-scale structure of Universe - galaxies: active - galaxies: statistics - radio continuum: galaxies - surveys
In striking contrast with the extremely high level of isotropy observed in the temperature of the cosmic microwave background (see e.g. de Bernardis et al. 2000), galaxies are not distributed throughout the Universe in a random manner. According to the gravitational theory of instability the present structures originated from tiny fluctuations in the initial mass density field. This has shaped the large-scale structure of the Universe, which consists of vast empty regions (voids), and strings of dark and luminous matter (walls) where billions of galaxies are found.
The clustering properties of galaxies can be quantified using statistical techniques, such as methods of nearest neighbour, counts in cells, power spectra, and correlation functions (see Peebles 1980 for an in-depth mathematical review). In particular the two-point correlation function is a simple, but powerful tool that has become a standard for studying large-scale structure. The clustering of cosmological objects can be characterized by their spatial correlation function, which has the form where r0 is the present-day correlation length and for objects ranging from clusters to normal galaxies (see Bahcall & Soneira 1983, for a review). The local population of galaxies is a relatively unbiased tracer of the underlying matter distribution, with r0=5.4 h-1 Mpc derived from galaxies in the early CfA redshift survey by Davis & Peebles (1983), however more recent low-redshift surveys show that the clustering of galaxies depends strongly on luminosity and/or morphological type. For example, local ellipticals represent spatial structures that are much more strongly clustered with h-1 Mpc (e.g. Norberg et al. 2002; Willmer et al. 1998; Guzzo et al. 1997). From deep, magnitude-limited redshift samples it has been found that the comoving correlation length of galaxies declines with redshift, roughly as expected from simple gravitational theory (e.g. CFRS, Le Fèvre et al. 1996; Hawaii K, Carlberg et al. 1997; CNOC2, Carlberg et al. 2000; CFDF, McCracken et al. 2001). In contrast to this, the clustering strength of quasars appears to vary little over . Croom et al. (2001) found an approximately constant amplitude of 5 h-1 Mpc from 10 000 quasars in the 2dF QSO Redshift Survey. Likewise, Daddi et al. (2001,2002)found that the (comoving) correlation length of massive elliptical galaxies also shows little evolution with redshift. They find h-1 Mpc for a population of extremely red objects (EROs) at (see also Roche et al. 2002; Firth et al. 2002; McCarthy et al. 2001), which are consistent with being the passively evolving progenitors of local massive ellipticals (e.g. Dey et al. 1999; Liu et al. 2000; Cimatti et al. 2002; Dunlop et al. 1996; Cimatti et al. 1998). Color selection methods such as Lyman-break (Steidel et al. 1995) and narrow-band imaging techniques are providing statistical samples of very high redshift galaxies, allowing us to study large-scale structure at even earlier epochs. Lyman-break galaxies have correlation lengths as high as h-1 Mpc even at , and are thought to be associated with (mildly) biased star-forming galaxies (e.g. Porciani & Giavalisco 2002; Adelberger 2000; Ouchi et al. 2001).
Studying clustering as a function of redshift and galaxy type may provide important constraints on some long-standing problems in cosmology concerning galaxy formation and evolution. For example, which of the galaxies observed at high redshift are the progenitors of local galaxy populations, and which of the local galaxies host the remnant black holes that once powered high redshift active galactic nuclei (AGN)? Two common views on how structures observed at high redshifts may be related to structures observed today are represented by (i) the galaxy conservation model (e.g. Tegmark & Peebles 1998; Fry 1996) in which it is assumed that galaxies formed very early in a monolithic collapse (e.g. Eggen et al. 1962) and have evolved passively with a decreasing star formation rate since , and (ii) the hierarchical merging model (e.g. Mo & White 1996) in which it is assumed that the most luminous galaxies formed more recently in massive dark matter haloes that have grown hierarchically by the merging of less massive galaxies and their haloes. Kauffmann & Charlot (1998) computed the evolution of the observed K-band luminosity function for both the monolithic case and the hierarchical case, and found that by a redshift of 1 these models differ greatly in the abundance of bright galaxies they predict. Likewise, the validity of these models can be tested by comparing predictions for galaxy clustering from numerical simulations or (semi-)analythic theory (e.g. Kauffmann et al. 1999b; Moustakas & Somerville 2002; Mo & White 2002, and references therein) with the observed clustering of a population of galaxies. In the case of pure monolithic collapse galaxy clustering is dictated by the evolution of galaxy bias under the rules of gravitational perturbation theory, but without the extra non-linear effects arising from galaxy mergers. Such a scenario can be thought of as a baseline model for the clustering of the matter as probed by galaxies situated in average mass haloes. However, in the hierarchical case the evolution of galaxy bias is much more complex, since galaxies are no longer conserved quantitities (Kauffmann et al. 1999b). Comparing their observations to model predictions Daddi et al. (2001) find that such a scenario best explains the clustering evolution of massive ellipticals out to .
Radio surveys can make an important contribution to this study: the use of magnitude-limited surveys for finding high redshift objects is usually a cumbersome task, while any flux density limited sample of radio sources contains objects at redshifts of (Dunlop & Peacock 1990). Powerful extra-galactic radio sources, or AGN in general, result from the fuelling of a supermassive blackhole (e.g. Rees 1990,1984), and there is evidence that the host galaxies of these high-redshift AGN are associated with some of the most massive structures in the early Universe (e.g. Crawford & Fabian 1996; Röttgering et al. 1996; Pentericci et al. 1999; Venemans et al. 2002; Best et al. 1998; McCarthy 1988). Moreover, because powerful AGN were far more numerous at than today, radio surveys can be used to probe a population of massive galaxies in the epoch of galaxy formation.
Despite initial concerns that any cosmological clustering of radio sources may be undetectable due to the relatively broad redshift distribution washing out the signal (e.g. Griffith 1993; Webster & Pearson 1977), Kooiman et al. (1995) detected strong clustering of bright radio sources in the 4.85 GHz 87GB survey. Cress et al. (1996) made a thorough analysis of clustering at the mJy-level. Using the 1.4 GHz FIRST survey (see also Magliocchetti et al. 1998) they obtained the first high-significance measurement of clustering from a deep radio sample, allowing them to investigate the separate contributions of both AGN and starburst galaxies (but see Wilman et al. 2003). Further results on the statistics of radio source clustering have been presented by Loan et al. (1997) and Rengelink (1998), who based their analysis on the 4.85 GHz Parkes-MIT-NRAO survey and the 325 MHz WENNS survey, respectively. In high-resolution surveys such as FIRST, large radio sources can become resolved in several components, thereby spuriously contributing to the cosmological clustering signal. Cress et al. (1996) and Magliocchetti et al. (1998) outlined the basic steps involved in separating the signal due to this effect from the true cosmological clustering, although the angular size distribution of radio sources at the mJy level is still largely unconstrained.
Since the individual redshifts of the radio sources are generally not known, one usually only measures the two-dimensional clustering by means of the angular correlation function, . However, the redshift distribution of the survey can be used to constrain r0. Using this so-called Limber inversion technique (Peebles 1980; Limber 1953; Rubin 1954; Phillipps et al. 1978), radio sources from the above surveys are typically found to have h-1 Mpc. Rengelink (1998) and Rengelink & Röttgering (1999) pointed out that this broad range in r0 measured can be explained by a scenario in which powerful radio sources have a larger r0 than less powerful radio sources. This would be highly consistent with the mounting evidence that powerful radio galaxies are the high-redshift progenitors of local cD-galaxies residing in massive environments that are hence strongly clustered. Here, we will further explore the hypothesis of Rengelink et al. by investigating the clustering of radio sources in a number of flux-limited subsamples taken from the 1.4 GHz NRAO VLA Sky Survey (Overzier 2001; see also Blake & Wall 2002a,b), the largest existing 1.4 GHz survey to date, containing radio sources down to a flux density limit of 2.5 mJy at (FWHM) resolution (Condon et al. 1998). We also present new results on clustering using the latest release of the FIRST survey, carefully taking into account the contribution of multiple-component radio sources, which we found to be severely underestimated in earlier analyses.
The outline of this article is as follows: in Sect. 2 we describe our methods for measuring the angular two-point correlation function. In Sect. 3 we describe the NVSS and FIRST radio surveys, and in Sect. 4 we present measurements of the angular clustering of the sources in these surveys and construct a simple model of the angular size distribution of radio sources. We derive an estimate of r0 as a function of flux density limit in Sect. 5. In Sect. 6 we compare our results with the results found for other populations of galaxies taken from literature, and discuss how the combined measurements relate to current theories on galaxy formation and evolution. The main conclusions are summarized in Sect. 7.
The galaxy angular two-point correlation function, ,
is defined as the excess
probability, over that expected for a Poissonian distribution, of finding a galaxy
at an angular distance
from a given other galaxy (e.g. Peebles 1980):
(1) |
(2) |
The NRAO VLA Sky Survey (NVSS) is the largest radio survey that currently exists at 1.4 GHz. It was constructed between 1993 and 1998 (Condon et al. 1998), and covers 10.3 sr of the sky north of (82% of the sky). Figure 1 indicates the coverage of the NVSS. With a limiting flux density of 2.5 mJy ( ) and an angular resolution of (FWHM), the NVSS contains about sources, and is considered to be 99% complete at a flux density limit of 3.4 mJy (Condon et al. 1998). The NVSS is based on 217 446 snapshot observations (of mostly 23 s) using the VLA in D- and DnC-configuration. These snapshots were then combined to produce a set of datacubes containing Stokes I, Q, and U images. A source catalogue was extracted by fitting the images with multiple elliptical Gaussians. Since the angular resolution of the NVSS ( FWHM) is well above the median angular size of extra-galactic radio sources ( arcsec), most sources in the catalogue are unresolved (95% for 3<S1.4<10 mJy). The main NVSS data products have been made publicly available for the use of the astronomical community, and can be obtained from the NRAO website.
RA (J2000) | DEC (J2000) | Remark |
- | - | Missing snapshot |
- | - | Missing snapshot |
- | - | Missing snapshot |
- | - | Missing snapshot |
- | - | Missing snapshot |
- | - | Missing snapshot |
- | - | 3C 48 |
- | - | Perseus A |
- | - | Fornax A |
- | - | 3C 123 |
- | - | PKS 0521-36 |
- | - | M 42 |
- | - | 3C 147.1 |
- | - | 3C 147 |
- | - | TXS 0549-051 |
- | - | 3C 173.1 |
- | - | Hydra A |
- | - | NGC 4261 |
- | - | 3C 273 |
- | - | M 87 |
- | - | MRC 1309-216 |
- | - | NGC 5127 |
- | - | 3C 295 |
- | - | Hercules A |
- | - | 3C 353 |
- | - | 3C 380 |
- | - | TXS 1921-293 |
Figure 1: Aitoff map of the NVSS source density. Scales run from below (black) to above the mean source density (white). The region of the galactic plane with is indicated by solid lines. Besides the expected enhancement of the source density due to the large population of galactic radio sources, the NVSS catalogue suffers from large numbers of spurious sources around bright or extended sources (white regions), as well as an overall decrease in the source density below (see the greyscale change at ). See text and Table 1 for details. |
Figure 3: Aitoff map of the FIRST source density. Scales run from below (black) to above (white) the mean source density. The region of the galactic plane with is indicated by solid lines. |
To optimize our catalogue for measuring the true cosmological clustering of radio sources, we have carried out a detailed examination of the NVSS source catalogue to identify and correct regions that may spuriously contribute to :
NVSS | FIRST | ||||
Region | Sources | Region | Sources | ||
3 mJy | , | 210 530 | 3 mJy | and | 188 885 |
and | |||||
5 mJy | , | 507 608 | 5 mJy | " " | 124 974 |
7 mJy | , | 351 079 | 7 mJy | " " | 94 099 |
10 mJy | 433 951 | 10 mJy | " " | 68 560 | |
20 mJy | 242 599 | ||||
30 mJy | 165 459 | ||||
40 mJy | 123 769 | ||||
50 mJy | 97 753 | ||||
60 mJy | 79 738 | ||||
80 mJy | 56 903 | ||||
100 mJy | 43 294 | ||||
200 mJy | 17 015 |
The FIRST (Faint Images of the Radio Sky at Twenty centimeters) survey (Becker et al. 1995) is another 1.4 GHz VLA survey, which was started in 1993 and is still under construction. Using the VLA in B-configuration it will ultimately cover 10 000 square degrees of the northern Galactic cap, matching the survey area of the Sloan Digital Sky Survey. Given the large coverage of FIRST, its sensitivity is unprecedented: with a limiting flux density of 1 mJy ( ) and an angular resolution of (FWHM) the catalogue contains about 100 sources per square degree with a completeness level of 95% at 2 mJy (Becker et al. 1995).
We have obtained the publicly available 2001 October 15 version of the source catalogue, which has been derived from the 1993 through 2001 observations, and covers about 8565 square degrees of the sky. About 4% of the 771 076 sources in the catalogue are flagged as possible side-lobes, which we exclude from the catalogue. We set the lower flux density limit of the catalogue to 3 mJy, the limiting flux density of the NVSS survey. Finally, we select the regions and , and from the catalogue, by requiring a relatively uniform source density and a simple geometric form. This area covers 5538 square degrees and contains 188 885 sources. As for the NVSS, we construct a map of the FIRST surface density (Fig. 3). and plot the source density as a function of declination (Fig. 5). For the selected region we found no suspicious features in the catalogue. The number of sources in various FIRST subsamples are listed in Table 2.
Figure 4: The rms-noise level as a function of galactic latitude. The average rms-noise level of the survey is 0.48 mJy beam-1. Dotted lines enclose the region . |
Figure 5: The FIRST source density as a function of declination for various limiting flux densities. |
Figure 6: The angular two-point correlation function of S>10 mJy NVSS sources. The power-law fits described in the text are indicated. |
Following the procedures described in Sect. 2 we compute for the S>10 mJy NVSS subsample. Distances between data and/or random positions are initially measured in bins of , and rebinned in bins of constant logarithmic spacing to analyse the data. We fit the data using a weighted -minimization routine, and we determine the errors from the covariance matrix.
The results are shown in Fig. 6. We find that two power-laws are needed to describe the full range of our measurements. Fitting the data with a power-law angular correlation function (e.g. Peebles 1980) at angular scales of gives a slope of , while at we find a slope of . The latter value is consistent with the slope of the empirical power-law of found for the cosmological clustering of objects ranging from normal galaxies to clusters (see Bahcall & Soneira 1983, for a review). However, at small angular scales the power-law is much steeper, presumably caused by the enhancement of due to the decomposition of large radio galaxies into their separate radio components (see Sects. 4.2 and 4.4; see also Blake & Wall 2002a). If we fit the data simultaneously with a double power-law correlation function of the form with fixed slopes of and , we find amplitudes of and . The double power-law fit is indicated in Fig. 6.
Although the median angular size of radio sources is (e.g. Condon et al. 1998), radio sources can have sizes of up to several arcminutes. At angular scales comparable to the size of these large radio galaxies, the true cosmological can become confused or even dominated by resolving these galaxies into their various radio components, such as lobes, hot spots and cores. The angular scale at which the size distribution of radio galaxies begins to dominate is indicated by the clear break around . Earlier studies attempted to correct for the contribution of multi-component radio sources by means of component combining algorithms. For example, Cress et al. (1996) calculated the angular correlation function for the FIRST survey considering all sources within of each other as a single source. The analysis of the FIRST data was repeated by Magliocchetti et al. (1998), who removed double sources using an algorithm based on the relation of Oort et al. (1987) and flux ratio statistics of the components of genuine doubles. They found values of , and for flux density limits between 3 and 10 mJy. Comparing their results to our measurement for the NVSS presented in Fig. 6, we conclude that despite the efforts of these authors it is likely that a residual contribution from large radio galaxies remained. Fitting the data over the whole range of with a single power-law explains the apparently high value of reported for the clustering of FIRST radio sources.
Here, we present new measurements from the FIRST survey. Our reasons for repeating the work of Cress et al. (1996) and Magliocchetti et al. (1998) are threefold.
Firstly, the FIRST catalogue has almost doubled in size, enabling a better statistical measure of .
Secondly, the clear break found in the
angular correlation function of the NVSS enabled us to isolate the signal due to true clustering from
the signal due to the size distribution of radio galaxies. A similar analysis can be applied
to the FIRST data. Thirdly, we found large-scale gradients in
the NVSS source density below a flux density limit of 10 mJy (see Sect. 3.2).
The FIRST data can be used to verify and complement the results from the NVSS for 3-10 mJy.
Figure 7: The angular two-point correlation function of S>3 mJy FIRST sources. The power-law fits described in the text are indicated. |
In Fig. 7 we present our measurements for the angular correlation function from the S>3 mJy FIRST subsample. As for the NVSS, we see a clear break in due to the presence of multi-component radio sources. Fitting the measurements with our double power-law model yields and , and and . Note that the break in in this sample occurs at compared to for S>10 mJy in NVSS (see Fig. 6). Blake & Wall (2002a) show that this is due to a dependency ( being the surface density of radio sources) of the amplitude of at small angular scales, simply because the weight of pair-counts due to large radio galaxies increases as the surface density decreases (see their Eq. (4)).
We conclude that the cosmological of S>10 mJy NVSS sources and S>3 mJy FIRST sources,
as determined by our analysis, are consistent with having the canonical clustering power-law slope
of
,
and an amplitude of
.
Figure 9: The amplitude of the cosmological angular correlation function ( ) of NVSS and FIRST as a function of 1.4 GHz flux density limit. For comparison, we have indicated the results for the WENSS and GB6 surveys from Rengelink (1998) and Rengelink & Röttgering (1999). |
NVSS | FIRST | |||
3 mJy | ||||
5 mJy | ||||
7 mJy | ||||
10 mJy | ||||
20 mJy | - | |||
30 mJy | - | |||
40 mJy | - | |||
50 mJy | - | |||
60 mJy | - | |||
80 mJy | - | |||
100 mJy | - | |||
200 mJy | - |
We would like to make the following remarks:
The steepening of the slope of at small angular scales is presumably related to multi-component sources spuriously enhancing the true clustering pair counts at small . To demonstrate the reality of this assumption, we create a simple model for the angular size distribution of radio galaxies in the NVSS, that is able to account for this extra signal contributing to . We model the physical size distribution of sources in our S>10 mJy NVSS sample, and use their redshift distribution to obtain the angular size distribution. Because we know the angular resolution of the NVSS, this model can then be used to estimate the fraction of sources likely to be resolved. It is essential to separate sources that are resolved into a single, elongated object from sources that are resolved into a number of components, since only the latter would produce extra pair counts. Here, we assume that the majority of surplus pair counts arise from resolving the two edge-brightened radio lobes of FRII-type radio galaxies (see Fanaroff & Riley 1974), and we estimate that the fraction of FRIIs at 10 mJy is 40% from Wall & Jackson (1997) (assuming a spectral index of to extrapolate to 1.4 GHz).
Several groups have investigated the median physical sizes of FRII radio galaxies as a function of
redshift and radio luminosity by parameterizing the linear size as
,
where P is the radio luminosity (for a review see Blundell et al. 1999). Results using different
samples of radio galaxies vary from no size evolution at all
(e.g. Nilsson et al. 1993), to size evolution depending only on redshift (e.g. Kapahi et al. 1987), and
size evolution depending on both redshift and luminosity with contradictory results
(e.g. Oort et al. 1987; Barthel & Miley 1988; Singal 1993). We use the results of Neeser et al. (1995) who found the following
linear size-redshift relation from a spectroscopically complete sample of FRII radio galaxies:
For the purpose of our model, we place simulated sources in small redshift intervals ( ) in the range , and assume that their mean physical size evolves with redshift according to Eq. (3). We set the total number of input sources equal to the estimated number of S>10 mJy FRIIs in our NVSS sample (40% of 434 000), and calculate the number of sources in each redshift interval from the redshift distribution, N(z), using the formalism of Dunlop & Peacock (1990) (see Sect. 5 for details). We then assume that in each redshift interval sizes are normally distributed. We take a mean size of 500 kpc and a standard deviation of 250 kpc at , chosen so that the resulting physical size distribution roughly resembles the distribution of projected linear sizes versus redshift as it is given by Blundell et al. (1999) for three complete samples of FRII radio galaxies from the 3C, 6C, and 7C radio surveys. The resulting physical size distribution is shown in Fig. 10, where we plot filled contours of the source density in the linear size-redshift plane to illustrate the underlying redshift distribution. We have also indicated the minimum physical size that is theoretically required for a source to become resolved as a function of redshift, given by the NVSS resolution of 45 (FWHM). We would like to remark at this point that the distribution of sizes in our model beyond redshifts of should not be taken too seriously as it is based on a straight extrapolation from measurements made at redshifts , and does not take into account the fact that at these high redshifts most sources will be extremely young and are thus likely to be very small. However, as can be seen from Fig. 10, our modeled size distribution falls below the NVSS resolution already at . Taking smaller sizes at higher redshifts will have no effect on the modeled size distribution of resolved sources that we want to derive here.
Assuming
we calculate the
angular size distribution associated with our model. We construct 10 such models, and average them
to get our final model of the angular size distribution of the sample.
This model is presented in Fig. 11. Although the mean angular size is 10
in agreement with
Condon et al. (1998), sizes are found to extend up to several arcminutes beyond the resolution
of the NVSS (indicated by the dotted line).
We now compare the number of surplus pairs expected from resolved FRII sources in the model,
,
to the actually measured pair counts at angular scales of
.
At these scales, the measured pair counts consist of both pair counts due to clustering and pair counts due to doubles, so
(4) |
(5) |
Several remarks that can be made are the following:
Figure 12: The ratio of observed doubles to modeled doubles per distance interval. The angular resolution of the NVSS is indicated by the dotted line. |
Figure 13: Dashed lines show the redshift distributions for S1.4>10 mJy, computed from the free-form models 1-4, the pure luminosity evolution model (PLE) and the luminosity/density evolution model (LDE) of Dunlop & Peacock (1990) (see text for details). The average of the six different models is indicated by the solid curve. |
At the mJy level and higher it is standard practice to compute redshift distributions
using the Dunlop & Peacock (1990) radio luminosity functions (RLFs). These authors have constructed a range of model luminosity functions using spectroscopically complete samples from several radio surveys
at different frequencies. Using a free-form modelling approach they found a number of smooth functions that were consistent with
the data. In addition, they attempted two models of a more physical nature by assuming pure luminosity evolution (PLE) and
luminosity/density evolution (LDE) to describe the RLF. The total ensemble is expected to agree well at those
luminosities and frequencies at which they are best constrained by the data, while uncertainties in the extrapolation
of each of these models to those regions that are less constrained by the data may be reduced by taking the ensemble as
a whole. We compute redshift distributions, N(z), for each flux-limited subsample using the free-form models 1-4 and
the PLE/LDE models for the combined population of flat (,
)
and steep (
)
spectrum radio sources given by Dunlop & Peacock (1990, taking the MEAN-z data from their appendix C) from
(6) | |||
Figures 13 and 14 show the redshift distributions for S>10 mJy and S>100 mJy, respectively. We calculate the average of the six different models (indicated by the solid curve), which will be our best estimate of N(z) use in the analysis below (the same method was used for the N(z) applied to the model of the angular size distribution described in Sect. 4.4). It is important to keep in mind that the functional form of N(z) remains virtually unchanged from 3-200 mJy. Over this range in flux densities the RLFs represent a broad redshift distribution with a peak around , indicating the very large median redshift that is generally probed by radio surveys.
Given the amplitudes of determined in Sect. 4 we can use the cosmological Limber equation to estimate
the spatial correlation length, r0, by deprojecting into the spatial
correlation function,
using the redshift distribution and cosmology (e.g. Peebles 1980, Chapt. 56).
We consider two cosmological models:
a flat, vacuum dominated, low-density Universe (CDM;
,
), and an Einstein-de Sitter model Universe (CDM;
,
).
We use H0=100 h km s-1 Mpc-1.
Figure 14: The redshift distributions for S1.4>100 mJy. See the caption of Fig. 13 for details. |
CDM | CDM | CDM | CDM | |
r0 (h-1 Mpc) | r0 (h-1 Mpc) | r0 (h-1 Mpc) | r0 (h-1 Mpc) | |
3 mJy | ||||
5 mJy | ||||
7 mJy | ||||
10 mJy | ||||
20 mJy | ||||
30 mJy | ||||
40 mJy | ||||
50 mJy | ||||
60 mJy | ||||
80 mJy | ||||
100 mJy | ||||
200 mJy |
We assume an epoch dependent power-law spatial correlation function of the form
(7) |
(8) |
(10) |
Q(z) | = | (11) | |
x(z) | = | ||
= |
The evolution parameter can represent a variety of clustering models. Three important cases are the following (see Kundic 1997; Phillipps et al. 1978). (1) The stable clustering model ( ): if galaxy clustering is gravitationally bound at small scales, then clusters have fixed physical sizes (i.e. they will neither contract nor expand) and will have a correlation function that decreases with redshift as (1+z)-1.2. (2) The comoving clustering model ( ): galaxies and clusters expand with the Universe, so their correlation function remains unchanged in comoving coordinates. This case applies well to a low density Universe where there is not enough gravitational pull to counterbalance expansion, and implies that structures have formed very early. (3) The linear growth model ( ): clustering grows as expected under linear perturbation theory.
Studies of the spatial clustering properties of radio-quiet quasars indicate that the clustering history of active galaxies, unlike that of normal galaxies, is best characterized using a negative value for . Kundic (1997) measured the high-redshift quasar-quasar correlation function from the Palomar Transit Grism Survey, and found no evidence for a decrease in the correlation amplitude of quasars with redshift. Moreover, he found that , suggesting an even higher amplitude at higher redshifts. Similarly, Croom et al. (2001) find almost no evolution in clustering strength for quasars taken from the 2dF QSO Redshift Survey out to . Therefore, we opt for evolution model 2 (i.e. constant clustering in comoving coordinates), which implies for . In Table 4 we list the results obtained using this model for the two different cosmological models. For comparison, we also indicate the results using the stable clustering model ( ). For the present-day correlation length is higher than for in both cosmologies. However, given the strong peak in the redshift distribution at , we are effectively measuring clustering at . Calculating in the case of stable clustering using Eq. (9) yields a value that is only lower than in the case of . Therefore, the value of is relatively independent of the exact value of . The results for the (CDM) case are presented in Fig. 15. We find an approximately constant spatial correlation length of 6.0 h-1 Mpc from 3-40 mJy, compared to 14 h-1 Mpc at 200 mJy.
As we have shown, the possibility that the observed flux-dependency of the clustering is just an effect of projection can be ruled out, since the shape of the redshift distribution is relatively constant with flux over several orders of magnitude (at least above 1 mJy). This automatically implies that the average radio power of the subsamples increases with flux density (indicated by the top axis of Fig. 15). An alternative explanation was therefore suggested by Rengelink (1998) and Rengelink & Röttgering (1999) based on their measurements of the clustering of radio sources in the WENSS and GB6 surveys. They concluded that the clustering signal could change as a function of flux density if relatively low and high power radio galaxies represent different spatial structures at a similar epoch (). Taking the predicted population mix of radio sources from Wall & Jackson (1997), we find that for S1.4>10 mJy the fractions of FRIs and FRIIs are about equal. However, for S1.4>100 mJy the fraction of FRIIs is more than 75%. Given the fractional changes of the source populations with flux density limit, the clustering amplitudes measured are very well matched by a scenario in which the clustering of powerful radio sources (mostly FRII) and average power radio sources (FRI/FRII) are intrinsically different, with FRIIs being more strongly clustered at than the radio galaxy population on average.
As pointed out by Rengelink (1998) and Rengelink & Röttgering (1999) the large difference in observing frequencies and sensitivities
of WENSS and GB6 (the limiting 1.4 GHz flux densities probed by these surveys correspond
to 10 mJy for WENSS and 70 mJy for Greenbank, respectively) only allowed them to make a comparison between the results, whereas
the detection of the inferred flux-dependency of r0 within a single survey would be highly desirable.
Our analysis of the clustering in the single large-area, intermediate-frequency NVSS survey is in agreement with their conclusions.
We start this section by making a survey of other clustering measurements from literature. However, readers may wish to skip directly to Sect. 6.2 for a discussion on these measurements and the results presented in this paper in their cosmological context.
In order to compare results from different studies, all values taken from literature were converted assuming a fixed slope by setting . All correlation lengths are expressed in comoving units, and we have transformed all values to a CDM cosmology (see Magliocchetti et al. 2000). Please note that the list given below is not complete, and the reader is kindly invited to consult the individual papers and the references therein for further information.
Estimates of the correlation length of rich Abell clusters are given by Bahcall & Soneira (1983) and Postman et al. (1992) who found h-1 Mpc. Lahav et al. (1989) found h-1 Mpc from an all-sky sample of the brightest X-ray clusters, and Dalton et al. (1994) and Croft et al. (1997) found h-1 Mpc and h-1 Mpc, respectively, for clusters selected from the APM Galaxy Survey. Recently, Gonzalez et al. (2002) measured the correlation length of distant clusters in the Las Campanas Distant Cluster Survey and found a correlation length of h-1 Mpc at .
Different studies may have sampled clusters of different degrees of richness, which can account for most of the scatter in the reported values. In general, however, all results are consistent with clusters being the most strongly clustered objects known in the Universe.
Bright early-type galaxies are found to have a strongly clustered distribution in the local Universe. Willmer et al. (1998) find h-1 Mpc for local ellipticals, and Guzzo et al. (1997) measure a considerably higher h-1 Mpc for a sample of similar galaxies. Although these results are only consistent with each other at the level, the latter sample contains a higher fraction of local clusters, presumably responsible for boosting the r0. The dependence of galaxy clustering on luminosity and spectral type has been studied using the ongoing 2 degree Field Galaxy Redshift Survey (2dFGRS). Norberg et al. (2002) find h-1 Mpc for the brightest early-type galaxies in the 2dFGRS. Moreover, they find a strong dependence of clustering strength on luminosity, with the amplitude increasing by a factor of 2.5 between L* and 4L*. The ordinary population of galaxies has been found to be less strongly clustered than the population consisting of local (bright) ellipticals: Loveday et al. (1995) find h-1 Mpc from the APM survey. At higher redshifts, the clustering strength in a sample of faint K-selected galaxies with minimum rest-frame luminosities of MK=-23.5, or about 0.5L*, is found to be fairly rapidly declining with redshift: Carlberg et al. (1997) find h-1 Mpc, h-1 Mpc, h-1 Mpc, and h-1 Mpc, at , , , and , respectively. Carlberg et al. (2000) present measurements on a sample of galaxies up to and find a much milder decline from h-1 Mpc at to h-1 Mpc at .
Clustering of the local population of IRAS-selected galaxies is best fit by h-1 Mpc (Fisher et al. 1994).
Several recent studies indicate that the comoving correlation length of early-type galaxies undergoes little or no evolution from . Evidence for this is provided by the clustering of extremely red objects, a population of galaxies having very red optical to infrared colors ( ). These red colors are consistent with them being either old, passively evolving elliptical galaxies, or strongly dust-enshrouded starburst galaxies at . Indeed, further observations have confirmed that both classes are present in the ERO population (e.g. Dey et al. 1999; Liu et al. 2000; Cimatti et al. 1998; Dunlop et al. 1996). Daddi et al. (2001) have recently embarked on a study of the spatial clustering of a large sample of EROs at , and found a large correlation length of h-1 Mpc. In Cimatti et al. (2002) the results are presented involving the EROs that were identified in a large flux limited redshift survey of 500 galaxies with . The derived fraction of early-type EROs from that sample is %, while there is an increasing contribution of dusty star-forming EROs at faint magnitudes. Therefore, Daddi et al. (2002) have attempted to analyse separately the spatial clustering of EROs from both categories by studying the frequency of close pairs. They find that the comoving correlation length of the dust-enshrouded starbursts is constrained to be less than r0=2.5 h-1 Mpc, while the old EROs are clustered with h-1 Mpc. This is consistent with the value reported earlier in Daddi et al. (2001), which is still valid as a lower limit for the clustering of early-type EROs based on the argument that the much less clustered dusty star-forming EROs only dilute the clustering signal coming from the ellipticals in this sample (see also Roche et al. 2002). Furthermore, McCarthy et al. (2001) have identified a large sample of such faint red galaxies as being consistent with mildly evolved early-type galaxies at . They find a clustering strength of h-1 Mpc.
The results on the spatial clustering of radio sources at presented in this paper indicate that r0 depends on radio luminosity in such a way that very luminous (FRII) radio galaxies cluster more strongly than the total population of radio galaxies (both FRI and FRII) on average, reminiscent of a similar luminosity trend found for samples of optically-selected galaxies. We roughly construct two radio luminosity bins from our measurements by comparing the r0 found for 3-40 mJy to the r0 found for the 200 mJy subsample. We find h-1 Mpc-1 for the relatively low power bin ( W Hz-1 sr-1), and h-1 Mpc-1 for the high power bin (P>1026 W Hz-1 sr-1).
Croom et al. (2001) have determined the correlation length of quasars (QSOs) using 10 558 quasars taken from the 2dF QSO Redshift Survey. They find that QSO clustering appears to vary little with redshift, with h-1 Mpc at , h-1 Mpc at , h-1 Mpc at , h-1 Mpc at , and h-1 Mpc at .
Lyman-break galaxies (LBGs) are found to be associated with star-forming galaxies at , with comoving correlation lengths of h-1 Mpc (Adelberger 2000), and h-1 Mpc (Porciani & Giavalisco 2002). Ouchi et al. (2001) find h-1 Mpc for a sample of LBGs at .
Figure 16: The redshift evolution of galaxy clustering in a CDM Universe. See the text for references to data taken from literature. Lines represent the following models: (i) stable clustering ( ) normalized to r0 of local ellipticals and clusters ( dotted lines), (ii) linear clustering ( ) normalized to 5 h-1 Mpc ( dot-dashed line), (iii) clustering of the dark matter ( thick solid line, from Jenkins et al. 1998, see also Moustakas & Somerville 2002 for a useful parameterization), (iv) galaxy conservation model normalized to r0 of local ellipticals ( thin solid line, see Fry 1996), (v) hierarchical model for clustering evolution of early-type galaxies normalized to r0 of local ellipticals ( thick dashed line, from Kauffmann et al. 1999b), and (vi) clustering evolution as a function of dark matter halo masses with ( thin dashed lines, from Matarrese et al. 1997). A nice representation of this figure showing actual images of the various objects rather than symbols can be found at our website: http://www.strw.leidenuniv.nl/~overzier/r0.html. |
In Fig. 16 we present an overview of the evolution of galaxy clustering, as it follows from the broad variety of observational results summarized above. The r0 that we measure for the brightest radio sources at is comparable to the r0 measured for bright ellipticals locally, and higher than the r0 measured for relatively faint radio sources and quasars, suggesting that they are considerably more biased and probably probe spatial structures associated with strongly clustered, massive objects. This does not come totally unexpectedly, as there is a range of observational evidence in support of this result. Best et al. (1998) found that powerful 3CR radio galaxies are mostly associated with massive galaxies at , and at high () and very high () redshifts the most luminous (i.e. FRII-type) radio sources are found in very dense environments associated with forming clusters. This is based on for example the presence of large X-ray halos (Crawford & Fabian 1996), excesses of companion galaxies (Nakata et al. 2001; Röttgering et al. 1996; McCarthy 1988), and excesses of Ly emitters around powerful radio sources (Kurk et al. 2000; Venemans et al. 2002). Furthermore, most very high redshift radio galaxies (z>2) are surrounded by giant halos of emission line gas (e.g. De Breuck et al. 2000; Röttgering et al. 1999), and some have very clumpy morphologies suggestive of massive forming systems (e.g. Pentericci et al. 2000,1999). Using HST/NICMOS observations, Pentericci et al. (2001) have found a number of radio galaxies at having morphologies that are represented well by a de Vaucouleurs profile, consistent with them being elliptical galaxies or proto-galaxy bulges.
As argued by Best et al. (1999), powerful radio sources must rely on (i) a plentiful supply of gas to fuel a supermassive blackhole that can drive the AGN activity, and (ii) a dense surrounding medium able to contain the radio lobes. These environments are indeed expected to be found in the gas-rich galaxy clusters at high redshift, additionally supporting the conclusion that high redshift FRIIs are associated with strongly clustered, massive objects. One may argue that this conclusion somewhat contradicts the fact that low redshift FRIIs are primarily found to be situated in small, isolated galaxy groups, and not in the centers of large clusters (Butcher & Oemler 1978; Hill & Lilly 1991). This, however, can easily be explained by considering that the local analogs of the gas-rich cluster environments that are suitable for producing powerful FRIIs at high redshifts, are found in relatively small galaxy groups, and not in the gas-depleted centers of local rich clusters (Rengelink 1998).
Interestingly, we find that both EROs and powerful radio galaxies are strongly clustered with h-1 Mpc at . Willott et al. (2001) suggested that high-redshift radio galaxies and EROs could be identical galaxies seen at different stages of their evolution, based on their findings of ERO-like host galaxies for a number of radio galaxies from the 7C Redshift Survey. This, of course, would be highly consistent with the belief that both radio galaxies and EROs may be the progenitors of local bright ellipticals. They conclude that the density of radio sources with minimum radio luminosities of log 10P151=24 W Hz-1 sr-1 is consistent with a model in which all EROs go through a relatively short period of AGN activity, forming a radio galaxy somewhere between z=2 and z=1.
However, if all EROs are radio galaxies at some stage, their highly clustered spatial distribution should be reflected in the spatial distribution of the radio galaxies. Figure 16 shows that the clustering of EROs and radio galaxies is consistent only for those galaxies with radio luminosities of log W Hz-1 sr-1. The surface density of such radio sources in the redshift range 1<z<2 in the NVSS is arcmin-2, while the surface density of EROs having and is 0.5 arcmin-2 (Daddi et al. 2001). If we take the fraction of old ellipticals among EROs to be 70% (Cimatti et al. 2002), then only 0.06% of these EROs are currently observed in their radio-loud phase. However, because the typically assumed AGN lifetimes are short compared to the cosmological time-scale from z=2 and z=1 ( Gyr for , ), the number of EROs that could undergo a radio-loud phase is 2-20% (assuming yr.). These fractions can be increased significantly if, for example, we select EROs that are much redder: the density of EROs having is a factor of 10 lower compared to (Daddi et al. 2001), giving 14-140%. It may be clear from the above that the unification of EROs and radio galaxies, although tempting, relies on a number of issues that have not yet been resolved. Further study of the luminosities, colors and morphologies of radio galaxy hosts, as well as the cluster environments of EROs may be expected to provide important clues for constraining this scenario.
Linear ( , dot-dashed line) or stable ( , dotted line) clustering evolution models have been found to best fit the measurements of ordinary, optically-selected galaxies at (e.g. Carlberg et al. 2000,1997; McCracken et al. 2001, and references therein). However, as Fig. 16 shows, these models do not provide a good description for the evolution of massive early type galaxies as inferred from the measurements of local bright ellipticals and FRII radio galaxies and EROs at . Adjusting these models to the measurements would either require massive ellipticals to have a correlation length around 6-7 h-1 Mpc, or local bright ellipticals to have a correlation length of the order of that of local clusters, far greater than observed. For these galaxies, the current measurements require a model that predicts relatively constant clustering in comoving coordinates, i.e. a negative value of in the simple -model.
Although the parameterization of clustering evolution by means of the -model is useful for characterizing the measurements as a function of redshift,
it does not provide good physical insight into evolution governed by the clustering of
dark matter halos (see McCracken et al. 2001; Giavalisco et al. 1998). Galaxy clustering evolution can be described more precisely
by
In the galaxy conservation model, objects are formed by means of monolithic collapse
at arbitrarily high redshift, and their clustering evolution is described solely by the cosmological growth of density
perturbations (Fry 1996). In this model, bias evolves as
b(z)=1+(b0-1)/D(z), | (13) |
Crucial to the picture that is developing may be the recent results of Wilson (2003), who studied the clustering of (V-I)-selected L* early-type galaxies in the redshift range 0.2<z<0.9. This author found that these galaxies cluster slightly more strongly compared to the field, with a best-fitting -model of and h-1 Mpc. This is in agreement with the correlation length of local L* early-types in the 2dFGRS. Wilson (2003) remarks that this measurement is inconstent with the large r0 found for EROs, which are also believed to be early-type galaxies. The value of r0 for EROs and radio galaxies could be spuriously high due to uncertainties in their redshift distributions which is not included in the quoted errors, although the selection functions of both EROs and powerful radio galaxies are considered to be understood relatively well (e.g. Dunlop & Peacock 1990; Daddi et al. 2001; McCarthy et al. 2001). Alternatively, EROs and radio galaxies at may be much more strongly clustered because they correspond to a population of massive, bright cluster galaxies in the process of formation. If FRII radio galaxies and EROs are indeed the distant analogs of local early-types, they are becoming considerably more biased tracers of the underlying galaxy distribution with redshift, while this galaxy distribution itself probably traces the dark matter distribution with relatively constant bias. Interestingly, (semi-) analytic models and N-body simulations are able to explain this bias evolution and the large inferred r0 at of massive ellipticals, if the assumption that galaxies are conserved quantities (i.e. closed-box systems) is relaxed. These hierarchical merging models (e.g. Kauffmann et al. 1999b; Mo & White 1996; Moustakas & Somerville 2002; Mo & White 2002; Matarrese et al. 1997; Moscardini et al. 1998, and references therein) prescribe that for certain types of objects bias can grow stronger with redshift than the growth of perturbations, resulting in a r0 that is constant or even increasing with redshift.
In the (transient) model of Matarrese et al. (1997) it is assumed that the mass of the dark matter halo also determines the
physical parameters of the galaxy that it contains. Based on the work of Mo & White (1996) and the formalism of Press & Schechter (1974),
Matarrese et al. (1997) derive that the bias in such a model evolves as
(14) |
In Fig. 16 we have also indicated the predicted evolution of the clustering of early-type galaxies (thick dashed line) from the CDM-models of Kauffmann et al. (1999a,b) (see also Somerville et al. 2001), normalised to r0 found for local ellipticals. An important feature of the models presented in Kauffmann et al. (1999b) is that one naturally expects a dip in r0 between z=0 and , if structure is probed by galaxies of intermediate luminosities residing in haloes of masses that have formed early and are unbiased tracers of the overall mass distribution. However, these simulations also show that this dip is very sensitive to sample selection criteria: massive early-type galaxies exhibit no dip in clustering between z=0 and , because they occur in rare, very masssive haloes of which are strongly biased locally, and which become even stronger biased with redshift. The agreement of this model with the results presented in this paper and the results of Daddi et al. (2001) and McCarthy et al. (2001) is striking. Although promising, some discrepancies between the model and the observations remain. For instance, Daddi et al. (2001,2002) find strong disagreement between the model and the high observed space density of EROs, seemingly consistent with the purely passive evolution of local ellipticals. Furthermore, current merging models generally predict that these galaxies should have experienced recent star-formation activity, while this is not observed. It may become possible to still reconcile the observations with the CDM merging models if, for example, the merging is accompanied by little star-formation (Daddi et al. 2001). Also, the EROs are found to have relatively old stellar populations of 3 Gyr that show no indications of recent formation processes. However, Moustakas & Somerville (2002) point out that the relatively old ages of their stellar populations do not automatically imply similar ages for the host galaxies.
Despite the success of current hierarchical models in predicting the evolution of bias for these massive galaxies, we would like to point out that galaxy conservation or linear/stable clustering evolution could still be able to explain the measurements if EROs and/or powerful radio galaxies are solely found in rich Abell-type clusters with (present-day) h-1 Mpc. As we have shown there is substantial evidence that this may be the case for, at least, the powerful radio galaxies, and future data may show whether this also holds for (a subset of) the population of EROs.
At the highest redshifts, clustering of LBGs at indicate that these objects can be connected to local ellipticals in a galaxy conservation scenario. However, it is now believed that LBGs probably occupy much less massive halos of than those that contain local massive galaxies, suggesting that if these objects are to be the progenitors of local ellipticals, they must have accumulated a considerable amount of mass (Moustakas & Somerville 2002; Adelberger 2000).
Figure 16 suggests that the clustering evolution of active galaxies in general is considerably different from that of ordinary galaxies. Albeit at a lower amplitude, the clustering of QSOs also shows a trend of constant or slightly increasing amplitude with redshift, very similar to the trend that we derive for the clustering of the most massive ellipticals. According to the standard paradigm, AGN are powered by the accretion of matter onto a (super-)massive blackhole (e.g. Rees 1984). This fuelling mechanism may very well be associated with the injection and accretion of gas during major merging events, and thus, the occurrence of AGN seems to be logically linked to the hierarchical scenarios for structure formation. Recently, in a series of papers (Haehnelt et al. 1998; Haehnelt & Kauffmann 2000; Kauffmann & Haehnelt 2000,2002) the simulations of Kauffmann et al. (1999b) were extended to a unified model for the evolution of both galaxies and quasars. In their model, elliptical galaxies, supermassive black holes and starbursts are formed during major merging events, in which a fraction of the available gas is used to trigger quasar activity by accretion for about 107 years, and the remaining gas is converted into stars in a single short burst. This model succesfully reproduces the evolution of cold gas that is derived from observations of damped Ly systems, the luminosity functions and clustering properties of QSOs from the 2dF QSO survey, and the relation between bulge velocity dispersion and black hole mass that has been found in demographic studies of black holes in nearby galaxies (e.g. Magorrian et al. 1998; Gebhardt et al. 2000; Kormendy & Richstone 1995).
Although it has yet remained unknown exactly what processes cause the physical differences between radio-quiet and radio-loud AGN, recent results indicate that the hosts of all powerful AGN (both radio-loud and radio-quiet) are almost exclusively ellipticals (see Dunlop & McLure 2003, and references therein). However, the same studies also indicate that while radio-quiet AGN hosts can have black holes with masses of , the radio-loud sources are cleanly confined to black hole masses . Furthermore, in the regime of extreme radio luminosities that lie well beyond the FRI/FRII luminosity-break, the power needed can only be achieved by blackholes with , requiring host masses of > that imply L>L* luminosities (Dunlop & McLure 2003). This may explain why the most powerful NVSS sources are extremely clustered compared to the, on average, less massive hosts of QSOs. This is supported by the fact that the radio sources in our lower radio luminosity bin have a correlation length similar to that of QSOs at , while both populations are still clustered more strongly compared to the field at . We conclude that the masses of the haloes, host galaxies, and black holes that are probed by the most powerful radio sources are among the most massive objects in the Universe, possibly formed through massive mergers in hierarchical fashion.
The main conclusions that can be drawn from our analysis are the following:
Acknowledgements
We would like to thank Chris Blake, Emanuele Daddi, Matt Jarvis, Melanie Johnston-Hollitt and Jaron Kurk for productive discussions and reading through the text. We also thank the referee for very helpful comments.