A&A 475, 83-99 (2007)
DOI: 10.1051/0004-6361:20077506

The spatial clustering of mid-IR selected star forming galaxies at z $\sim$ 1 in the GOODS fields

R. Gilli¹ - E. Daddi² - R. Chary³ - M. Dickinson⁴ - D. Elbaz² - M. Giavalisco⁵ - M. Kitzbichler⁶ - D. Stern⁷ - E. Vanzella⁸

1 - Istituto Nazionale di Astrofisica (INAF) - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy
2 - Laboratoire AIM, CEA/DSM - CNRS - Université Paris Diderot, DAPNIA/SAp, Orme des Merisiers, 91191 Gif-sur-Yvette, France
3 - Spitzer Science Center, California Institute of Technology, Mail Stop 220-6, Pasadena, CA 91125, USA
4 - National Optical Astronomy Observatory, PO Box 26732, Tucson, AZ 85726, USA
5 - University of Massachusetts, Astronomy Dept, Amherst, MA 01003, USA
6 - Max-Planck Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, 85748 Garching b. München, Germany
7 - Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
8 - Istituto Nazionale di Astrofisica (INAF) - Osservatorio Astronomico di Trieste, via G. Tiepolo 11, 34131 Trieste, Italy

Received 19 March 2007 / Accepted 27 July 2007

Abstract
We present the first spatial clustering measurements of $z\sim 1$, 24 $\mu$m-selected, star forming galaxies in the Great Observatories Origins Deep Survey (GOODS). The sample under investigation includes 495 objects in GOODS-South and 811 objects in GOODS-North selected down to flux densities of $f_{24}>20~\mu$Jy and $z_{\rm AB}<23.5$ mag, for which spectroscopic redshifts are available. The median redshift, IR luminosity and star formation rate (SFR) of the sample are $z\sim 0.8$, $L_{\rm IR}\sim4.4\times10^{10}$ $L_{\odot}$, and ${\it SFR}\sim 7.6~M_{\odot}$ yr^-1, respectively. We measure the projected correlation function $w(r_{\rm p})$ on scales of $r_{\rm p}=0.06{-}10~h^{-1}$ Mpc, from which we derive a best fit comoving correlation length of $r_0 = 4.0\pm0.4~h^{-1}$ Mpc and slope of $\gamma=1.5\pm0.1$ for the whole $f_{24}>20~\mu$Jy sample after combining the two fields. We find indications of a larger correlation length for objects of higher luminosity, with Luminous Infrared Galaxies (LIRGs, $L_{\rm IR}>10^{11}~L_{\odot}$) reaching $r_0\sim 5.1~h^{-1}$ Mpc. This would imply that galaxies with larger SFRs are hosted in progressively more massive halos, reaching minimum halo masses of $\sim$ $3\times10^{12}~M_\odot$ for LIRGs. We compare our measurements with the predictions from semi-analytic models based on the Millennium simulation. The variance in the models is used to estimate the errors in our GOODS clustering measurements, which are dominated by cosmic variance. The measurements from the two GOODS fields are found to be consistent within the errors. On scales of the GOODS fields, the real sources appear more strongly clustered than objects in the Millennium-simulation based catalogs, if the selection function is applied consistently. This suggests that star formation at $z\sim0.5{-}1$ is being hosted in more massive halos and denser environments than currently predicted by galaxy formation models. Mid-IR selected sources appear also to be more strongly clustered than optically selected ones at similar redshifts in deep surveys like the DEEP2 Galaxy Redshift Survey and the VIMOS-VLT Deep Survey (VVDS), although the significance of this result is $\la$$3\sigma$ when accounting for cosmic variance. We find that LIRGs at $z\sim 1$ are consistent with being the direct descendants of Lyman Break Galaxies and UV-selected galaxies at $z\sim2{-}3$, both in term of number densities and clustering properties, which would suggest long lasting star-formation activity in galaxies over cosmological timescales. The local descendants of $z\sim0.5{-}1$ star forming galaxies are not luminous IR galaxies but are more likely to be normal, L<L_* ellipticals and bright spirals.

Key words: galaxies: evolution - cosmology: large-scale structure of Universe - cosmology: observations

  
1 Introduction

In the general paradigm of large scale structure formation, the small primordial fluctuations in the matter density field progressively grow through gravitational collapse, leading to the present-day complex network of clumps and filaments which is often referred to as the "Cosmic Web''. Baryons are believed to cool within dark matter halos (DMHs) and form galaxies and cluster of galaxies, whose distribution on the sky should then trace that of the underlying dark matter. While the formation and the evolution of dark matter structures can be followed in a relatively straightforward way through N-body simulations (e.g., Jenkins et al. 1998; Springel et al. 2005), which can be also approximated analytically with high accuracy (Peacock & Dodds 1996), the physics of baryon cooling and galaxy formation within DMHs is far more complex. As a result of these complex physical processes, the distribution of galaxies in the sky may be biased with respect to that of the underlying matter distribution. The amplitude of this bias is expected to evolve with cosmic time and be dependent on galaxy type, luminosity and local environment (Norberg et al. 2002). The comparison between the clustering properties of galaxies and those of DMHs predicted by cold dark matter (CDM) models can be used to evaluate the typical mass of the DMHs in which galaxies form and reside as a function of cosmic time. Following the evolution of the typical DMH hosting a given galaxy type at any given time also allows one to predict the environment in which that galaxy should be found nowadays and the environment in which it was residing in the past. In other words, under reasonable assumptions, it is possible to guess the progenitors and descendants of galaxy populations observed at any cosmological epoch.

Galaxy clustering has been traditionally studied by means of the two-point correlation function $\xi(r)$, defined as the excess probability over random of finding a pair of galaxies at a separation r from one another, which is often approximated with a power law of the form $\xi(r)=(r/r_0)^{-\gamma}$. In the local Universe different clustering properties have been observed as a function of galaxy type. In the Sloan Digital Sky Survey (SDSS, York et al. 2000), at a median redshift of $z\sim0.1$, red, early-type galaxies show a larger correlation length and a steeper slope ( $r_0=6.8~h^{-1}~{\rm Mpc ,}\:\gamma=1.9$) than blue, late type galaxies ( $r_0=4.0~h^{-1}~{\rm Mpc ,}\:\gamma=1.4$; Zehavi et al. 2002). Similar results arise from the 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001), in which, at a similar median redshift, passive galaxies show a correlation length and slope of $r_0=6.0~h^{-1}~{\rm Mpc ,}\:\gamma=1.9$ as opposed to $r_0=4.1~h^{-1}~\rm {Mpc ,}\:\gamma=1.5$measured for star forming galaxies (Madgwick et al. 2003).

At cosmologically significant distances, deep surveys on sky patches of less than 1 deg², complemented by large spectroscopic campaigns, are measuring the clustering of high redshift objects with reasonable accuracy. The separation between the clustering properties of star forming and passively evolving galaxies seems to be well established even at redshifts around 1. In the DEEP2 Galaxy Redshift Survey, Coil et al. (2004) found that $z\sim 0.9$ galaxies with absorption line spectra have a correlation length significantly larger than emission-line galaxies at the same redshift. A similar result has been found in the VIMOS-VLT Deep Survey (VVDS) by Meneux et al. (2006), who measured a correlation length that was larger for red galaxies than for blue galaxies at $z\sim 0.8$.

Porciani & Giavalisco (2002) and Adelberger et al. (2005) measured the clustering properties of star forming galaxies selected by the Lyman-break technique between redshifts of 1.7 and 3 (see also Hamana et al. 2004; Ouchi et al. 2005; and Lee et al. 2006, for Lyman break galaxies selected at $z\sim4{-}5$). The measured comoving correlation length of 4.0-4.5 h^-1 Mpc for these high redshift objects is expected to increase with time and suggests that they will evolve into moderate-luminosity, elliptical galaxies by z=0 (Adelberger et al. 2005).

While all of the above described samples are based on optical selection, star formation in galaxies can be efficiently traced by mid-infrared observations. The star formation rate (SFR), particularly the dust obscured component, is indeed directly correlated to the mid-IR luminosity, which is in turn a robust proxy for the total (8-1000 $\mu$m) IR luminosity (e.g., Spinoglio et al. 1995; Chary & Elbaz 2001; Forster-Schreiber et al. 2004). This has been demonstrated in the present-day Universe, but seems to hold at least up to $z\sim 1$, where the bulk of star-formation occurs in dust-obscured regions. Indeed, the deepest existing radio data have shown that $L_{\rm IR}$ values determined from the mid-IR luminosity of galaxies at $z\sim 1$ are consistent with those derived using the radio to IR luminosity correlation (Elbaz et al. 2002; Appleton et al. 2004).

Early work by the Infrared Astronomical Satellite (IRAS) showed that the correlation length of nearby (median $z\sim0.03$) mid-IR bright galaxies ( $f_{60~\rm\mu m}>1.2$ Jy) is about 4 h^-1 Mpc (Fisher et al. 1994), in agreement with the values measured for local star forming objects by the SDSS and 2dFGRS. More recently, an attempt to measure the clustering properties of mid-IR selected sources at fainter flux densities has been made (D'Elia et al. 2005). Based on a small sample of galaxies detected by the Infrared Space Observatory (ISO) with $f_{15~\rm\mu m}>0.5$ mJy, D'Elia et al. found that the clustering level measured for these $z\sim 0.2$ galaxies is similar to that measured by IRAS for more local sources.

The Spitzer Space Telescope (Werner et al. 2004), with its unprecedented sensitivity at mid-IR and far-IR wavelengths, is enabling further progress to be made. Deep surveys at 24 $\mu$m are being carried out in different regions of the sky (see, e.g., Papovich et al. 2004), with the deepest ones being performed in the two GOODS fields down to $f_{24}\sim 10{-}20~\mu$Jy (Chary et al., in preparation). For the first time, this allows us to select field galaxies based on their ongoing level of star formation activity at a wavelength where dust corrections are negligible. This is a more physically motivated selection than those based on qualitative galaxy properties like color bi-modality. It thus provides greater insight into the nature of galaxy and star formation in the distant Universe and a more straightforward comparison to galaxy formation models. Our goal is to investigate the spatial distribution of $z\la 1$ star forming galaxies, and assess the relation between environment and star-formation rate. By constraining the nature of the descendants of star forming galaxies at $z\la 1$, we provide insight into the nature of downsizing of galaxy formation, a well established pattern for galaxy evolution which implies that star formation is taking place preferentially in more massive galaxies at higher redshifts (e.g., Cowie et al. 1996). A tight correlation between galaxy mass and star-formation rate has been discovered, with slope close to unity. This correlation has been shown to exist both in the local Universe as well as at $z\sim1.2$ (Noeske et al. 2007; Elbaz et al. 2007) with tentative evidence that it may be valid even at $z\sim 2$ (Daddi et al. 2007a). As more massive galaxies are on average hosted in more massive halos, we expect to find a direct correlation between clustering strength and star formation rate in the distant Universe.

Given the large (5-6 arcsec FWHM) resolution of the MIPS instrument (Multiband Imaging Photometer for Spitzer; Rieke et al. 2004) confusion problems due to blending are severe at the faintest flux densities. This makes a proper measure of the angular correlation function of faint MIPS sources difficult, leaving the 3D correlation function as the most viable method for estimating their clustering properties. In this paper, we measure the spatial clustering of 24 $\mu$m selected sources in the two GOODS fields by means of the projected correlation function  $w(r_{\rm p})$. Blending problems at short scales are completely overcome in this case, as angular clustering terms are negligible as discussed later in the paper.

The paper is organized as follows: in Sect. 2 we describe the data sets and the selection criteria adopted to define the samples used in the clustering analysis. In Sect. 3 we present the methods utilized to estimate the correlation function. In Sect. 4 several safety checks are performed to validate the adopted techniques. Simulations are also run to estimate errors on our measurements due to cosmic variance. The results of our analysis are presented in Sect. 5. In Sect. 6 the clustering measurements are discussed, interpreted and compared to estimates from optical surveys. The conclusions are presented in Sect. 7.

Throughout this paper, a flat cosmology with $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda}=0.7$ is assumed. Unless otherwise stated, we refer to comoving distances in units of h^-1 Mpc, where $H_0=100\;h$ km s^-1 Mpc^-1. Luminosities are calculated using h=0.7.

	Figure 1: Spectroscopic completeness down to $z_{\rm AB}=23.5$ mag for galaxies with $f_{24}>20~\mu$Jy in GOODS-N ( upper curve) and GOODS-S ( lower curve).
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2 The samples

The GOODS-South and GOODS-North fields, each covering about $10\times16$ arcmin, have been observed by Spitzer as part of the Great Observatories Origins Deep Survey Legacy Program (Dickinson et al. 2007, in preparation). Deep 24 $\mu$m observations with MIPS were carried out down to sensitivities of $\sim$$12~\mu$Jy ($\sim$$3\sigma$) in both fields (Chary et al., in preparation). Source catalogs at shorter wavelengths (Dickinson et al., in preparation) based on the Infrared Array Camera (IRAC; Fazio et al. 2004), were used as prior positions in order to improve source deblending and identify unique counterparts. Spectroscopic redshifts have been collected for about 60% of the MIPS sources with $z_{\rm AB}<23.5$ mag from a compilation of all the different follow-up spectroscopy programs carried out in the GOODS fields. In particular, for the GOODS-S field, we use the spectroscopic redshifts made available by Le Fevre et al. (2004), Mignoli et al. (2005), Vanzella et al. (2005, 2006). Redshifts in GOODS-N have been published in many papers over the years. At the redshifts of interest in this paper, the largest portion of the published redshifts can be found in Cohen et al. (2000), Wirth et al. (2004), and Cowie et al. (2004). We supplement these with additional redshifts for 24 $\mu$m selected sources from Stern et al. (in preparation). The spectroscopic completeness down to $z_{\rm AB}=23.5$ mag is shown in Fig. 1. In both fields the completeness level decreases towards fainter magnitudes, but in GOODS-N it is systematically higher than than in GOODS-S. For sources with $z_{\rm AB}<23.5$ mag the completeness level in GOODS-N is 65%, compared to 50% in GOODS-S. Only sources at 0.1<z<1.4 were considered in this work. The z<1.4 limit is imposed in order to remain in a redshift range where the spectroscopic sampling is highest, and where the observed 24 $\mu$m flux density can be used as an accurate tracer of the total IR luminosity of galaxies. Although $24~\mu$m observations can be used to obtain reasonable measurements of star formation activity averaged over the galaxy population at even higher redshifts (e.g., Daddi et al. 2005), individual sources with anomalous properties may show significant errors in their derived  $L_{\rm IR}$ (Daddi et al. 2007a; Papovich et al. 2007). Redshift quality flag information is available for most of the spectroscopic surveys done in GOODS-S, but is missing for some of the surveys in GOODS-N. In GOODS-S we considered only objects with high quality flags. In GOODS-N we have excluded some galaxies (<1% of the total sample) which appear to have incorrect spectroscopic redshifts, based on the shape of their spectral energy distribution and photometric redshifts. Furthermore, we have limited our analysis to sources with $f_{24}>20~\mu$Jy, for which the flux density estimate is reliable and source confusion is well understood (Chary 2006). About 20% of the sources fall below this limit and are therefore excluded from our clustering analysis. In total, 558 objects in GOODS-S and 875 objects in GOODS-N are found to satisfy these selection criteria (including AGN, see later). After accounting for spectroscopic incompleteness, the number of $f_{24}>20~\mu$Jy sources in GOODS-S and GOODS-N differ by $\sim$$20\%$. As shown in Sect. 5.3, this is consistent with being due to cosmic variance.

	Figure 2: 24 $\mu$m flux density vs redshift for sources detected by Spitzer/MIPS in GOODS-S (open circles) and GOODS-N (filled circles). Only sources with spectroscopic redshifts are shown. The dashed line shows the $f_{24}=20~\mu$Jy flux limit used in this work.
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	Figure 3: 24 $\mu$m flux density vs. $z_{\rm AB}$ magnitude for sources detected by Spitzer/MIPS in GOODS-S (open circles) and GOODS-N (filled circles). Only sources with spectroscopic redshifts are shown. The dashed line shows the $f_{24}=20~\mu$Jy flux limit used in this work.
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	Figure 4: $L_{\rm IR}$ vs. redshift for sources detected by Spitzer/MIPS in GOODS-S (open circles) and GOODS-N (filled circles). Only sources with spectroscopic redshifts are shown. The dashed line shows the $f_{24}=20~\mu$Jy flux limit used in this work.
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In Figs. 2 and 3 the 24 $\mu$m flux densities of sources in the two GOODS fields are plotted against their spectroscopic redshifts and $z_{\rm AB}$ magnitudes, respectively. Fainter 24 $\mu$m sources have on average fainter optical counterparts and tend to be at higher redshifts, although the redshift dependence of the average 24 $\mu$m flux density appears rather weak. Several redshift structures can be immediately identified, which are also traced by sources selected at other wavelengths (e.g., Cohen et al. 1996; Gilli et al. 2003; Barger et al. 2003). The 24 $\mu$m flux density and redshift distribution in the two fields are similar (see also Fig. 5). The median 24 $\mu$m flux density, optical magnitude and redshift for the considered samples are $f_{24}\sim 74~\mu$Jy, $z_{\rm AB}\sim21.8$ mag and $z\sim0.75$, respectively. We compute the total (8-1000 $\mu$m) IR luminosity $L_{\rm IR}$ from the observed 24 $\mu$m flux density, assuming the luminosity-dependent model templates of Chary & Elbaz (2001). The total IR luminosity provides a measure of the star formation rate in the galaxy using the relation ${\it SFR}=L_{\rm IR}\times 1.72\times10^{-10}~M_{\odot}$ yr^-1(Kennicutt et al. 1998). We note that if more recent estimates of the stellar initial mass function (IMF) are adopted (Kroupa 2001; Chabrier 2003), the same $L_{\rm IR}$ systematically converts into a $\sim$30% lower SFR. The exact conversion rate does not have an important effect on our results. The $L_{\rm IR}$ (SFR) versus redshift plot for the galaxy sample considered here is shown in Fig. 4, along with the $L_{\rm IR}$ cut introduced at each redshift by the $f_{24}>20~\mu$Jy selection. The luminosity distribution is similar in the two fields. The median luminosity and star formation rate are $4.4\times10^{10}\;L_{\odot}$ and 7.6 $M_{\odot}$ yr^-1, respectively. About 90% of the objects in the two fields have $L_{\rm IR}>10^{10}\;L_{\odot}$ while about 30% have $L_{\rm IR}>10^{11}\;L_{\odot}$. The latter are classified as Luminous Infrared Galaxies (LIRGs), and are forming stars at an average rate of $\sim$35 $M_{\odot}$ yr^-1.

	Figure 5: Redshift distribution for MIPS sources with $f_{24}>20~\mu$Jy and spectroscopic redshift in the GOODS-S ( left) and GOODS-N ( right) fields (AGN excluded), binned to $\Delta z = 0.01$. The smooth curves show the selection function adopted to generate the random control sample, obtained by smoothing the observed redshift distributions and truncated at z<0.1 and z>1.4 as the data samples.
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We note that the SFR estimated from the $L_{\rm IR}$ values may be a lower limit to the true galaxy SFR since it excludes the unobscured star-formation traced by the observed UV emission. We therefore considered B-band data from the Advanced Camera for Surveys (ACS) onboard the Hubble Space Telescope (HST), which traces the rest frame UV flux for galaxies at z>0.5, i.e., for the majority of the sources in our sample. We found that the SFR increases by only 4% on average when including the ACS data. We also note that the fraction of galaxies for which the SFR may have been underestimated significantly (e.g., by a factor of 1.5-2), is less that 4%. Due to the fact that the UV flux may have a contribution from old, evolved stars, these correction factors are upper limits. Our estimates appear to be in good agreement with those of Bell et al. (2005), who derive an average UV contribution of 5-10% to the global (mid-IR + UV) SFR of $z\sim 0.7$ star forming galaxies observed by Spitzer. Furthermore, since the UV correction decreases with increasing SFR, it becomes completely negligible for LIRGs. To summarize, UV corrections to the SFR do not have a significant impact on our results and are therefore neglected in the following analysis.

While most of these mid-IR selected sources are expected to be star forming galaxies (elliptical galaxies should be virtually absent from mid-IR selected samples), a significant fraction of sources may be active galactic nuclei (AGN), in which the radiation absorbed by circumnuclear material is re-emitted in the IR regime. Based on the X-ray properties of sources, we therefore tried to eliminate AGN interlopers. Both fields have been observed by Chandra with extremely deep (1-2 Msec) exposures (Giacconi et al. 2002; Alexander et al. 2003). Using an AGN classification similar to that adopted in Gilli et al. (2005), we flagged as AGN those sources with either observed 0.5-10 keV luminosities above 10⁴² erg s^-1 or with a column density above $N_{\rm H}=10^{22}$ cm^-2. The column density was estimated by assuming an intrinsic AGN template with spectral index of 0.7 and absorbing it at the source redshift to reproduce the observed hard-to-soft X-ray flux ratio. About 8% of the sources were removed from the samples using this AGN classification. We nonetheless verified that, due to the small fraction of AGN candidates, our results are insensitive to the methodology adopted to remove AGN. Moreover, our conclusions do not vary significant even if AGN are not excluded from the sample.

After the AGN are removed, we are left with samples of 495 and 811 galaxies, in GOODS-South and North, respectively. One may wonder if our samples are significantly contaminated by AGN which went undetected in the X-rays. Indeed, Alonso-Herrero et al. (2006) in GOODS-S and Donley et al. (2007) in GOODS-N, respectively, have identified a large population of IR luminous galaxies showing power-law emission in the IRAC $3.6{-}8~\mu$m bands. The power-law emission is thought to be due to hot dust in the vicinity of the AGN. Yet, half of these sources do not have an X-ray counterpart. We verified that none of these power-law AGN are present in our samples. We note that the Donley et al. (2007) and Alonso-Herrero et al. (2006) samples are based on shallow $24~\mu$m data, span a broader redshift range and primarily include objects with photometric redshifts. In contrast, our galaxies sample much fainter 24 $\mu$m flux densities and have spectroscopic redshifts of z<1.4. We are in the process of defining IR-based AGN samples in our deep MIPS catalogs. Preliminary analysis suggests that $\la$$10\%$ of sources might be flagged as additional AGN candidates and in principle should be removed from our samples. Their impact on the clustering measurements presented in this work is unlikely to be significant and will be discussed elsewhere when the AGN catalogs are finalized. Very recently, Daddi et al. (2007b) have shown that a population of highly obscured AGN, which are both undetected in the X-rays and do not show a power-law continuum in the IRAC bands, hide in about 20-30% of IR luminous ( $L_{\rm IR}\ga 10^{12}\;L_{\odot}$) galaxies at $z\sim 2$, providing a significant contribution to their 24 $\mu$m emission (see also Fiore et al. 2007). Given the relatively low IR luminosities ( $L_{\rm IR}\sim10^{10{-}11}~L_\odot$) and the longer mid-IR rest-frame wavelengths probed here at $z\sim 0.7$, we expect that the effect of contamination from an obscured AGN population will be less important for our study.

It should also be noted that we are measuring the clustering properties of mid-IR selected galaxies over a broad redshift range from z=0.1 to z=1.4. Star forming galaxies are undergoing rapid cosmological evolution in luminosity/density over this redshift range (e.g., Le Floc'h et al. 2005), and the clustering strength is also likely to evolve. Although most of the clustering signal measured in this work is due to galaxy pairs at $z\sim 0.7$, our measurements could be returning a value for the clustering strength that is an average between 0<z<1.4. Thus, our analysis is not identical to that obtained by considering an ideally large galaxy sample in a narrow redshift interval around $z\sim 0.7$. This caveat should be borne in mind when comparing our results with those obtained from other surveys.

3 Analysis techniques

To eliminate the distortions introduced by peculiar velocities and redshift errors, which affect the computation of the source clustering in redshift space, we resort to the projected correlation function, defined as in Davis & Peebles (1983):

$$w(r_{\rm p}) = \int_{-r_{v0}}^{r_{v0}} \xi(r_{\rm p}, r_v){\rm d}r_v,$$ (1)

where $\xi(r_{\rm p}, r_v)$ is the two-point correlation function expressed in terms of the separations perpendicular ($r_{\rm p}$) and parallel (r_v) to the line of sight, in comoving coordinates.

If the real space correlation function can be approximated by a power law of the form $\xi(r)=(r/r_0)^{-\gamma}$ and $r_{v0}= \infty$ then the following relation holds (Peebles 1980):

$$w(r_{\rm p}) =A(\gamma) r_0^{\gamma} r_{\rm p}^{1-\gamma},$$ (2)

where $A(\gamma)=\Gamma(1/2)\Gamma[(\gamma-1)/2]/\Gamma(\gamma/2)$ and $\Gamma(x)$ is Euler's Gamma function. $A(\gamma)$ increases from 3.68 when $\gamma=1.8$ to 7.96 when $\gamma=1.3$. The integration limit r_v0 is fixed to 10 h^-1 Mpc to maximize the correlation signal (see the end of this section).

To estimate the correlation function $\xi(r_{\rm p}, r_v)$ we used the Landy & Szalay (1993) estimator, which has been shown to have a nearly Poissonian variance and which appears to outperform other popular estimators (e.g., see Kerscher et al. 2000):

$$\xi(r_{\rm p}, r_v)=\frac{[DD]-2[DR]+[RR]}{[RR]}\cdot$$ (3)

[DD], [DR] and [RR] are the normalized data-data, data-random and random-random pairs, i.e.,

$$[DD]\equiv DD(r_{\rm p}, r_v)\frac{n_{\rm r}(n_{\rm r}-1)}{n_{\rm d}(n_{\rm d}-1)}$$ (4)

$$[DR]\equiv DR(r_{\rm p}, r_v)\frac{(n_{\rm r}-1)}{2 n_{\rm d}}$$ (5)

$$[RR]\equiv RR(r_{\rm p}, r_v),$$ (6)

while DD, DR and RR are the number of data-data, data-random and random-random pairs at separations $r_{\rm p} \pm \Delta r_{\rm p}$ and $r_v \pm \Delta r_v$; $n_{\rm d}$ and $n_{\rm r}$ are the total number of sources in the data and random sample, respectively.

Figure 6: Distribution of 24 $\mu$m sources with $f_{24}>20~\mu$Jy and with spectroscopic redshifts over the GOODS fields. GOODS-S is shown in the left panel while GOODS-N is shown in the right panel. The GOODS-S and GOODS-N fields are centered at (RA $_{\rm J2000}$, Dec $_{\rm J2000}$) = (53.122368, -27.797262) and (189.215744, 62.234791), respectively. Sources in the random samples have been placed at the coordinates of the real sources.

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In order to avoid confusion, we specify how galaxy pairs are counted. The number of DD and RR pairs have been counted only once, i.e., the total number of pairs in the real and in the random samples are $n_{\rm d}(n_{\rm d}-1)/2$ and $n_{\rm r}(n_{\rm r}-1)/2$, respectively. This accounts for the factor of 2 in the denominator of Eq. (5). This way of counting DD and RR pairs has been adopted by e.g., Landy & Szalay (1993), Guzzo et al. (1997), Gilli et al. (2005), Meneux et al. (2006). Other authors, instead, count DD and RR pairs $\it twice$, i.e., the total numbers of pairs in their real and random samples are $n_{\rm d}(n_{\rm d}-1)$ and $n_{\rm r}(n_{\rm r}-1)$, respectively, which removes the above mentioned factor of 2 from their formulae. These latter definitions have been adopted e.g., by Davis & Peebles (1983), Kerscher et al. (2000), Zehavi et al. (2002), Coil et al. (2004). It can be easily shown that the two formulations lead to the same $\xi(r)$. A simple way to see this is to replace DD and RR in the formulae of Zehavi et al. (2002) and Coil et al. (2004) with 2DD' and 2RR', where DD and RR are the numbers of pairs counted twice, while DD' and RR' are the numbers of pairs counted once ("independent'' pairs).

We note that in Eqs. (4) and (5), $n_{\rm d}$ is the number of sources observed in each GOODS field separately. Ideally, instead of using the observed number of sources, which may produce an overestimate (underestimate) of the clustering amplitude in under-dense (over-dense) regions, one should use the true mean source number, which is unknown. In principle, averaging the densities of the GOODS-N and GOODS-S fields would give a better approximation to the mean source density. However, because of the different spectroscopic completeness in the two GOODS fields, the estimate of the average density in a given redshift range may be non-trivial. One possibility is to assume that the total number of sources in the redshift range considered in this work ( z=0.1-1.4) is 20% larger in GOODS-N than in GOODS-S. This would be comparable to the difference observed in the total surface density of MIPS sources (after accounting for the 65% and 50% total spectroscopic completeness of GOODS-N and GOODS-S, respectively). However, since the spectroscopic completeness is a function of redshift and optical magnitude, and the completeness curves are different between the two fields (see Fig. 1), this may not be the case. At any rate, we have verified that, assuming that the z=0.1-1.4 source density is 20% larger in GOODS-N than in GOODS-S, the use of an averaged density value (i.e., increasing $n_{\rm d}$ by 10% in GOODS-S and decreasing it by the same amount in GOODS-N) gives a $\sim$$10\%$ shorter (longer) correlation length in GOODS-S (GOODS-N) than that estimated by using the density of each field separately. These fluctuations are of the same order as produced by redshift structures in our fields (see Sect. 5.1) and well within the cosmic variance errors (Sect. 4). Therefore, they do not change the main conclusions of the paper.

Since both the redshift and the coordinate $(\alpha, \delta)$distributions of the selected MIPS sources are potentially affected by observational biases, special care has to be taken in creating the sample of random sources. We adopted a procedure that has been shown to work well for X-ray AGN selected in the same fields (see Gilli et al. 2005). The redshifts of the random sources were extracted from a smoothed distribution of the real one, which should then include the same observational biases. We assumed a Gaussian smoothing length $\sigma_z = 0.2$ as a good compromise between smoothing scales that are too small (which suffer from significant fluctuations due to the observed redshift spikes) and scales that are too large (where on the contrary the source density of the smoothed distribution at a given redshift might not be a good estimate of the average observed value). For each of the source subsamples considered in this work (see Table 1), we smoothed the corresponding observed redshift distribution. The observed and smoothed redshift distributions for the $f_{24}>20~\mu$Jy samples are shown in Fig. 5. Due to the numerous redshift spikes observed, we did not try to measure the correlation function in different redshift bins since this would be extremely sensitive to the choice of bin boundaries. The coordinates ( $\alpha,\delta$) of the random sources were extracted from the coordinate ensemble of the real sample in order to reproduce the same uneven distribution on the plane of the sky. This procedure will in principle, reduce the correlation signal, since it removes the effects of angular clustering. However, as will be verified later, in deep, pencil-beam surveys like GOODS, where the radial coordinate spans a much broader distance than the transverse coordinate, most of the signal is due to redshift clustering, while angular clustering contributes at most a few percent. The distribution on the sky of the real sample is shown in Fig. 6. Each random sample is built to contain more than 10 000 sources.

The source pairs were binned in intervals of $\Delta{\rm log}~r_{\rm p}=0.1$, and $w(r_{\rm p})$ was measured in each bin. The resulting data points were then fit with a power law and the best fit parameters $\gamma$ and r₀ were determined via $\chi^2$ minimization. Given the small number of pairs which fall into certain bins (especially at the smallest scales), we used the formulae of Gehrels (1986) to estimate the 68% confidence interval (i.e., $1\sigma$ errorbars in Gaussian statistics).

It is well known that Poisson error bars underestimate the uncertainties in the correlation function when source pairs are not independent, which is the case for our sample. More importantly, these uncertainties do not account for cosmic variance. In the next section we assess the errors to be assigned to our best fit parameters by measuring $w(r_{\rm p})$on a series of simulated galaxy catalogs.

A practical integration limit r_v0 has to be chosen in Eq. (1) in order to maximize the correlation signal. Indeed, one should avoid r_v0 values which are too large since they would mainly add noise to the estimate of $w(r_{\rm p})$. On the other hand, scales which are too small, comparable to the redshift uncertainties and to the pairwise velocity dispersion, should also be avoided since they would not allow recovery of the entire signal. To search for the best integration limit r_v0, we measured $w(r_{\rm p})$ and the corresponding best fit r₀ and $\gamma$ values for different r_v0 values ranging from 3 to $100\:h^{-1}$ Mpc. Since deviations from a simple power law are sometimes observed (in particular for $r_{v0}=20{-}50\:h^{-1}$ Mpc in GOODS-N), using the best fit correlation length or clustering amplitude $A=r_0^{\gamma}$ as a measure of the clustering level is incorrect. To overcome this problem, we chose to quote the $w(r_{\rm p})$ values on a representative scale, as a function of r_v0. We adopt $r_{\rm p}=1$ h^-1 Mpc as our representative scale, since it is well within the considered $r_{\rm p}$ range, and is a separation at which the projected correlation function, $w(1\:h^{-1}$ Mpc), is determined with good accuracy.

In Fig. 7 we plot w(1 h^-1 Mpc) as a function of the radial integration limit r_v0. We note that the signal amplitude keeps increasing up to $r_{v0}\sim 10{-}20\:h^{-1}$ Mpc. For r_v0 values greater than $10{-}20\:h^{-1}$ Mpc, w(1 h^-1 Mpc) does not vary significantly. In the following, we therefore fix r_v0 to 10 h^-1 Mpc. Such a value for the integration limit is consistent with what has been widely used in the literature (e.g., Carlberg et al. 2000).

Figure 7: Projected correlation function $w(r_{\rm p})$ at $r_{\rm p}=1$h-1 Mpc measured in GOODS-S (open circles) and GOODS-N (filled circles) as a function of the integration limit rv0 (see Eq. (1)). Errorbars take into account cosmic variance (see Sect. 4). The turndown at very large scales in GOODS-S is likely due to sampling noise, in the regime where rv0 is much larger than the size of the redshift peaks (Gilli et al. 2003).

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4 Safety checks and error estimate

We have checked to see if our method for generating the random sample can bias in some way the best fit correlation parameters that we measure. In particular, placing the random sources at the coordinates of the real sources completely removes the contribution of angular clustering to the total clustering signal, which could bias the measured correlation length to lower values. We quantify this effect by considering 428 sources within a radius of 4.8 arcmin from the center of GOODS-N, where the spectroscopic coverage is most complete. We measured the correlation function in two ways: first, by placing the random sources at the coordinates of the real sources, and second, by placing the random sources truly at random within this area. When using this second method, r₀ increases by only 4%. Therefore, most of the clustering signal is provided by clustering along the radial direction, validating the adopted technique.

Another confirmation that this technique is not producing biased measures comes from tests performed on the mock galaxy catalogs based on the Millennium Simulation (Springel et al. 2005). These catalogs have been obtained by modeling galaxy formation through semi-analytic recipes applied to the pure dark matter N-body simulations of the Millennium run. Physical processes like gas cooling, star formation, supernovae and AGN feedback are taken into account, which are described in detail in Croton et al. (2006) and De Lucia & Blaizot (2007). Here we considered the most recent work by Kitzbichler & White (2007), who built a number of simulated light cones for deep galaxy surveys over 2 deg² sky fields. Each cone contains about $6.5\times10^5$ objects, for which a number of observable and physical properties like redshift, optical and near-IR magnitude, and star formation rate are listed. We considered one of these mock catalogs and applied to the simulated galaxies the same selection criteria adopted to define our data samples (see details in Sect. 2). Here some assumptions have to be made, since neither the $z_{\rm AB}$ magnitude, nor the 24 $\mu$m flux density are directly available for the simulated sources. We used $I_{\rm AB}$ as a proxy for $z_{\rm AB}$, assuming the I-z color expected for star forming galaxies at $z\sim 0.8$ ( $(I-z)_{\rm AB}=0.24$, Bruzual & Charlot 2003). Also, we converted the model star formation rate into IR luminosity using the relation ${\it SFR}=L_{\rm IR}\times 1.72\times10^{-10}~M_{\odot}$ yr^-1 and then, at each redshift, considered only objects above the $L_{\rm IR}$ threshold plotted in Fig. 4, which corresponds to the $f_{24}>20~\mu$Jy threshold used to define our data samples. The final mock sample contains about 50 000 objects, for which we computed the projected correlation function over the same $r_{\rm p}$ range used for the GOODS data, first placing the random control sources at the positions of the Millennium sources and then placing the random control sources really at random within the 2 deg² field. No significant variations are observed between the projected correlation function computed in the two cases, suggesting again that the contribution of angular clustering is negligible.

As shown in Table 1, when the same selection criteria are applied to the Millennium galaxies, these have on average different redshifts and luminosities than real mid-IR selected galaxies. We note however that our main goal is not to select mock galaxies with average properties identical to the real ones, but investigate any difference (e.g., in the average $L_{\rm IR}$ or SFR) between the data and the galaxy formation models once real and mock galaxies have been selected in the same way. This issue will be addressed in Sects. 5 and 6.

Table 1: Summary of the best fit clustering parameters. Poissonian uncertainties (only) are quoted here to allow comparison between different galaxy samples within the same GOODS field (see text). When comparing the results between the two fields, or when comparing the average properties of GOODS sources with those of other fields, cosmic variance uncertainties must also be included (see Table 2).

The mock catalogs from the Millennium simulation have also been used to estimate the global errors on the best fit parameters r₀ and $\gamma$, and to evaluate cosmic variance on the scale of the GOODS fields. This has been achieved by extracting from one of the Millennium mock catalogs samples of galaxies with progressively redder R-I colors and in the same redshift range as the GOODS galaxies. The clustering strength of the mock samples increases with redder R-I color threshold. We then split the $1.4\times1.4$ deg field over which each sample is distributed into 40 independent rectangles with the dimensions of a GOODS field (i.e., $10\times16$ arcmin). For each color sample, we measured the projected correlation function in each rectangle and computed the rms of the r₀ and $\gamma$ distributions. After subtracting in quadrature the (small) term due to Poissonian noise, we are left with the intrinsic cosmic variance. This procedure allows us to compute the appropriate variance for sources that are clustered similarly to the GOODS galaxies considered. We found that, on GOODS-sized fields, the fractional rms of the correlation length increases from 14% for sources with $r_0\sim 4\;h^{-1}$ Mpc to 20% for sources with $r_0=5.2\;h^{-1}$ Mpc, i.e., for populations as clustered as our total and LIRGs samples, respectively (see the next sections). Using the fractional rms values found with this method, the global errors related to our samples can be easily estimated once the Poissonian term is added back in quadrature. When averaging the properties of the two GOODS fields and presenting the results for the combined GOODS-S plus GOODS-N sample (see, e.g., Table 2), the variance estimated from the simulations is divided by a factor of $\sqrt 2$.

Table 2: Combined GOODS-S plus GOODS-N sample. The uncertainties take into account cosmic variance and have been computed as described in Sect. 4.

We note here that the error term due to cosmic variance should only be considered when comparing the clustering of the same population of sources across different fields, while it should be ignored when investigating clustering trends among different source sub-populations in the same field. Indeed, cosmic variance should increase or decrease the overall clustering amplitude over a given sky region, without modifying significantly the relative clustering between different galaxy subsamples (e.g., sources with different $L_{\rm IR}$), provided that their redshift distributions are similar, i.e., sources in the different subsamples are tracing the same large scale structures. For this reason, in Table 1 we quote only Poissonian uncertainties, suitable for comparison between different samples within the same field. When comparing the properties of the same population of sources between GOODS-N and GOODS-S, the cosmic variance term should be included. When this is done, we find that that the clustering amplitudes measured in the two fields are fully compatible with each other (see next section). In Table 2 we quote the clustering parameters averaged between the two samples, with uncertainties that include cosmic variance.

The Millennium mock catalogs, in which large source samples can be selected to minimize statistical noise, were also used to check if limiting the integration radius r_v0 to $10\:h^{-1}$ Mpc may introduce a systematic bias on our clustering measurements. We selected a population of mock galaxies with R-I>0.65, which shows a clustering level similar to that of our MIPS sources ( $r_0\sim 4\;h^{-1}$ Mpc), and measured $w(r_{\rm p})$ as a function of the integration radius r_v0. We found that for $r_{v0}=30\;h^{-1}$ Mpc the clustering signal already saturates, and we verified that for $r_{v0}=10\;h^{-1}$ Mpc the r₀ value is biased low by 5% with respect to the full, saturated value. In the Millennium catalogs, "purely cosmological'' redshifts are also available which are free from peculiar velocities. We used these to compute the correlation function in redshift space $\xi(r)$ for the same mock sample, which should provide an unbiased measurement of r₀. The resulting r₀is in very good agreement with that measured from  $w(r_{\rm p})$ for $r_{v0}\geq 30\;h^{-1}$ Mpc and therefore confirms that when using $r_{v0}=10\;h^{-1}$ Mpc, r₀ is biased low by 5%. We therefore conclude that the r₀ measurements presented in this work could underestimate the real values by $\sim$$5\%$. At any rate, we do not try to correct for this small systematic bias since it is found to be well within the uncertainties due to cosmic variance.

Finally, one may wonder if the fitting procedure to $w(r_{\rm p})$ adopted in the previous section, in which a simple Poisson weighting of the datapoints is used without considering the effects of cosmic variance, may bias the best fit parameters r₀ and $\gamma$. We verified that, when attributing to each $w(r_{\rm p})$ datapoint the cosmic variance error as a function of $r_{\rm p}$ resulting from our simulations, the best fit parameters r₀ and $\gamma$ are essentially unchanged. In the GOODS-N field r₀ and $\gamma$ change only by $\sim$$2\%$. In the GOODS-S field the change is smaller than 1%. This is due to the fact that the datapoints guiding the fits in both procedures are those with $r_{\rm p}$ in the range $0.5{-}4\;h^{-1}$ Mpc, which have both smaller Poisson errors and cosmic variance. In the following we will therefore keep using the fitting procedure described in Sect. 3.

	Figure 8: Projected correlation function measured for the total $f_{24}>20~\mu$Jy MIPS samples in GOODS-S and GOODS-N (open and filled circles, respectively) compared with that obtained from the Millennium simulation on a 2 deg2 field (filled triangles). Errorbars for the GOODS samples take into account cosmic variance (see Sect. 4). The best fit power laws are shown as dashed lines.
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5 Results

Having defined the analysis methods to estimate the galaxy projected correlation function and the global errors related to it, we are now in the position to measure the clustering properties of star forming galaxies in GOODS-S and GOODS-N and to compare them with those expected for mock galaxies from the Millennium simulation. Also, the clustering properties of different source subsamples, defined e.g., on the basis of their IR luminosity, can be readily investigated.

5.1 Correlation function of the full GOODS-S and GOODS-N samples

We first measured the projected correlation function for the total GOODS-N and GOODS-S samples over the projected scale range $r_{\rm p}=0.06{-}10\:h^{-1}$ Mpc. The results are shown in Fig. 8. In both fields a clear clustering signal is measured, with very high significance (>$35\sigma$). The best fit parameters (r₀, $\gamma$) are 4.25 h^-1 Mpc, 1.51 in GOODS-S and 3.81 h^-1 Mpc, 1.52 in GOODS-N (see Table 1). The clustering amplitude therefore appears about 10% larger in GOODS-S than in GOODS-N, confirming that the GOODS-S field has more structure than the GOODS-N field, as already noted from X-ray selected sources (Gilli et al. 2005). As shown in Fig. 8, most of the excess signal in GOODS-S is produced at projected scales in the range $0.8<r_{\rm p}<3\:h^{-1}$ Mpc, while at smaller and larger scales the signals measured in the two fields are almost identical. A simple check was performed by computing the projected correlation function in the GOODS-S field after removing those sources within the two redshift spikes at z=0.67 and z=0.73, which showed that most of the excess signal at $0.8<r_{\rm p}<3\:h^{-1}$ Mpc is indeed produced by these two structures. At any rate, as it will be shown later, the difference among the two r₀ values is fully accounted for by cosmic variance.

It should be noted that in the $r_{\rm p}=0.06{-}10\:h^{-1}$ Mpc scale range considered here, the datapoints at the smallest and largest scales are the least reliable. At small scales, e.g., $r_{\rm p}<0.1\:h^{-1}$ Mpc, source pairs at high redshifts (z>1.2) have separations on the plane of the sky comparable to the MIPS angular resolution at $24~\mu$m. Therefore source blending may be an issue. Furthermore, other biases might be introduced by the different angular selection functions of the many spectroscopic campaigns from which our catalogs are built. Also, the transverse size of the GOODS fields (19 arcmin diagonal) becomes smaller than $r_{\rm p}\sim 8\:h^{-1}$ Mpc for pairs at z<0.5. The corresponding  $w(r_{\rm p})$ measurements may thus be distorted with respect to those at smaller scales because of the different redshift range sampled. At any rate, the datapoints at the smallest and largest scales have the largest errorbars and thus do not significantly affect the overall estimate of the best fit parameters r₀ and $\gamma$. Indeed, when repeating the fits limiting the $r_{\rm p}$ range to  $0.1{-}8\:h^{-1}$ Mpc (or even $0.4{-}8\:h^{-1}$ Mpc), we obtained results in agreement with the previous ones within the errors. In the following computations we simply considered datapoints from $r_{\rm p}=10\:h^{-1}$ Mpc all the way down to the smallest scale from which we get signal.

At scales $r_{\rm p}\la 0.3\:h^{-1}$ Mpc, the correlation function data points appear to lay above the best fit power law, which may indicate that the intra-halo clustering term, i.e., the clustering term due to galaxy pairs within the same dark matter halo, is emerging, as has recently been seen in very large galaxy samples (e.g., SDSS, Zehavi et al. 2004). However, because of the possible biases in the $w(r_{\rm p})$ datapoints at smaller $r_{\rm p}$ scales mentioned above, the observed small-scale excess should be considered with caution. We will return to this in the Discussion.

The clustering behavior measured for the GOODS samples appears markedly different from the expectations from the Millennium simulation. As explained in the previous section, we computed the projected correlation function for a sample of about 50 000 objects in a mock galaxy catalog based on the Millennium run after applying the same selection criteria used for the real data. The projected correlation function for the mock catalog is also shown in Fig. 8 and the best fit clustering parameters are quoted in Table 1. Simulated mid-IR selected sources appear much less clustered than real sources. The overall $w(r_{\rm p})$ shape is also very different, with a flattening below 0.8 h^-1 Mpc, as opposed to the steepening observed in GOODS, and a steepening above $r_{\rm p}\sim 3{-}4$ h^-1 Mpc, whereas the GOODS $w(r_{\rm p})$ appears to have a constant slope.

A similar discrepancy between the predictions based on the Millennium mock catalogs and the real data has also been reported by McCracken et al. (2007), who measured the angular correlation function (ACF) of I-band selected galaxies in the COSMOS field. While at bright magnitudes the COSMOS and the Millennium ACF are in good agreement, at fainter magnitudes, I>22 mag, Millennium sources are less clustered than the real COSMOS sources, with an overall correlation function shape very similar to the one we measured for Millennium. In the same work, McCracken et al. (2007) point out that the observed discrepancy cannot be accounted for by cosmic variance.

	Figure 9: Best fit correlation length ( upper panel) and slope ( lower panel) measured over 40 mock fields obtained by splitting the 2 deg2Millennium field into independent rectangles with the dimensions of a GOODS field. The average r0 and $\gamma$ values (solid lines) and dispersion (shaded areas) are also shown.
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We checked to see if the discrepancy we find can be ascribed to cosmic variance by dividing the 2 deg² simulated mock field into 40 non-overlapping rectangles with the same size as that of the GOODS fields (i.e., $10\times16$ arcmin) and measuring average and dispersion of the r₀ and $\gamma$ distributions over these regions. As shown in Fig. 9, we found r₀=2.58, $\sigma_{r_0}=0.25$ for the average correlation length and its dispersion, and $\gamma=1.50$, $\sigma_{\gamma}=0.10$ for the average slope and its dispersion. Repeating this exercise on two other independent 2 deg² mock catalogs yielded similar results.

The correlation lengths measured in the GOODS-S and GOODS-N fields then appear to be about 6 and 5 standard deviations, respectively, larger than the value measured from the Millennium catalog. It therefore seems unlikely that the stronger clustering measured in the GOODS fields be produced by cosmic variance. Several possible explanations for this discrepancy are investigated in the Discussion, as well as a series of caveats that have to be kept in mind when comparing models with observations.

It is interesting to note how the average correlation length and slope measured on these $10\times16$ arcmin mock subsamples are smaller than those measured for the full 2 deg² mock catalog and reported in Table 1. One reason is that at large projected separations, where the Millennium $w(r_{\rm p})$ is steeper, the relative weight of the $w(r_{\rm p})$datapoints is much higher in the full 2 deg² field than in any GOODS-sized field, since distant galaxy pairs are much better sampled. As an example, over the whole $r_{\rm p}=0.06{-}10\;h^{-1}$ Mpc range considered in this work, the number of pairs in a typical GOODS-sized field is maximum in the range $r_{\rm p}=3{-}6\;h^{-1}$ Mpc, while in the full 2 deg² field it steadily increases towards larger projected separations. Another reason may be related to the effects of the integral constraint (Groth & Peebles 1977), which bias the measurements of the correlation function on finite size fields. We estimate that the bias introduced by the integral constraint may affect the $w(r_{\rm p})$ estimates by at most a few percent at the largest scales probed here (above 5 h^-1 Mpc).

Figure 10: Left: projected correlation function for sources with $L_{\rm IR}>10^{10}~L_{\odot}$ ( ${\it SFR}>1.7\;M_{\odot}$ yr-1) as measured in GOODS-S, GOODS-N and Millennium simulation (open circles, filled circles and filled triangles, respectively). Errorbars for the GOODS samples take into account cosmic variance (see Sect. 4). The best fit power laws are shown as dashed lines. Right: as in the left panel but for LIRGs, i.e., objects with $L_{\rm IR}>10^{11}~L_{\odot}$ ( ${\it SFR}>17\;M_{\odot}$ yr-1).

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5.2 Dependence of clustering on IR-luminosity/star formation rate

Recent observations have shown that, among star-forming galaxies at any redshift, the star formation rate appears to be correlated with the galaxy mass (Noeske et al. 2007; Elbaz et al. 2007; Daddi et al. 2007a). This is in agreement with the predictions from semi-analytic models of structure formation (Finlator et al. 2006; Kitzbichler & White 2007), though models also predict that this correlation breaks down for the most massive galaxies. It is therefore interesting to investigate if and how the clustering of galaxies depends on the IR luminosity, which is a good proxy for the star formation rate. We measured the projected correlation function for sources with $L_{\rm IR}>10^{10}\;L_{\odot}$ and for LIRGs ( $L_{\rm IR}>10^{11}\;L_{\odot}$), as shown in Fig. 10. In both fields we measure an increase of the clustering level with IR luminosity, with r₀ going from $\sim$$4\:h^{-1}$ Mpc for the whole samples to $\sim$$5\:h^{-1}$ Mpc for the LIRGs (see also Tables 1 and 2). A comparison between the correlation length of the different samples is shown in Fig. 13 for the combined GOODS-S plus GOODS-N fields. Because of the unavoidable degeneracy between luminosity and redshift which characterizes any flux limited sample, LIRGs are on average at higher redshifts than the full IR galaxy population. However, as reported in Table 1, while the median luminosity of LIRGs is about a factor of 5 larger than that of the total sample, their median redshift of $z\sim 1.0$ is not dramatically higher than that of the total sample, z=0.75. The modest difference in the median redshift for the two samples suggests that luminosity, not cosmic time, is the main factor contributing to the clustering dependence that we observe. Because the dark matter clustering is smaller at higher redshift, the difference would be even larger for the implied galaxy bias. Since r₀ for a given galaxy population is expected to increase with time, i.e., towards lower redshifts (see Sect. 6.4), properly accounting for the redshift differences between subsamples would actually strengthen the detection of IR luminosity segregation of clustering.

In order to properly establish the statistical significance of the trend of clustering versus luminosity, we also considered sources with $L_{\rm IR}\leq 10^{11}\;L_{\odot}$ (non-LIRGs), which therefore constitute a source sample disjoint from the LIRGs (see Table 1). The difference between the clustering correlation length of LIRGs and non-LIRGs is about $3\sigma$ and $5\sigma$ significant in GOODS-S and GOODS-N, respectively. As explained in Sect. 4, only the Poissonian errorbars quoted in Table 1 have been considered for this estimate. However, since the redshift distributions of the LIRGs and non-LIRGs samples are rather different (e.g., the median redshift for LIRGs is $z\sim 1.0$, while for non-LIRGs it is $z \sim 0.6{-}0.7$; see Table 1), this evidence must be investigated further since the two populations might not be tracing the same large scale structures. We have therefore restricted our analysis to the redshift range z=0.5-1.0, which allows us to compare LIRGs and non-LIRGs at similar median redshifts (see Table 1). Figures 11 and 12 show the redshift distributions and the projected correlation functions $w(r_{\rm p})$ measured for the z=0.5-1.0 LIRGs and non-LIRGs in the GOODS-S and GOODS-N field, respectively. Because of the limited source statistics, we used larger $r_{\rm p}$ bins ( $\Delta{\rm log}~r_{\rm p}=0.2$) than those previously adopted, and limit our analysis to the $r_{\rm p}=0.4{-}8\;h^{-1}$ Mpc range, where the $w(r_{\rm p})$measure is more robust. We found that the significance of stronger clustering of LIRGs decreases slightly, to $\sim$ $2{-}4\sigma$, when performing this more appropriate comparison at similar median redshifts. Although the measured correlation lengths are quite sensitive to the choice of the redshift bin boundaries because of the spiky nature of the observed redshift distributions, we note that we systematically measure larger correlation lengths for LIRGs than for non-LIRGs, even adopting other redshift intervals. We conclude that our data suggest an increase of the correlation length with average $L_{\rm IR}$ or SFR, although this result needs to be confirmed using larger samples with better statistics.

As in the case of the total sample, we compared the results from the GOODS fields with those from the Millennium simulation. In Fig. 13 the r₀ values of the samples with $L_{\rm IR}>10^{10}~L_{\odot}$ and $L_{\rm IR}>10^{11}~L_{\odot}$ in the redshift range z=0.1-1.4 for the combined GOODS-S plus GOODS-N sample (see Table 2) are plotted as a function of the sample median luminosity and compared with the expectations from mock samples extracted from Millennium using the same $L_{\rm IR}$ thresholds. Since in each Millennium sample the median $L_{\rm IR}$ is lower than in the corresponding GOODS sample (see Table 1) - and this is especially true for LIRGs - we also measured $w(r_{\rm p})$ for mock sources above $2\times10^{11}\;L_{\odot}$, which have the same median luminosity of GOODS LIRGs. Again, we used the 40 GOODS-sized subregions of the 2 deg² full mock field to obtain the average correlation length and dispersion for model galaxies selected at different luminosities. This is shown by the shaded region in Fig. 13. Even at high luminosities, the overall clustering of the data appears stronger than that predicted by the simulations, although with reduced significance.

Figure 11: Upper panel: redshift distributions and selection functions for LIRGs and non-LIRGs in GOODS-S. Sources in the z=0.5-1.0 redshift interval used to compute the projected correlation function $w(r_{\rm p})$ shown in the lower panel have been shaded. Lower panel: projected correlation function $w(r_{\rm p})$measured in GOODS-S for LIRGs and non-LIRGs in the redshift interval z=0.5-1.0. Poisson errorbars are used here since the comparison is performed between samples with similar redshift distributions in the same field. The best fit power laws are shown as dashed lines.

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	Figure 12: Same as in Fig. 11 but for the GOODS-N field.
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As noted above, among galaxies with $f_{24}>20~\mu$Jy, the fraction of IR luminous objects is lower in the mock catalog than in GOODS. As an example, the fraction of LIRGs is 13% in Millenium, as opposed to the 30% in GOODS (see Table 1). This is related to the fact that, as emphasized by Elbaz et al. (2007), Millennium galaxies are forming stars at rates $\approx$3 times lower than those which are observed at $z\sim 1$. We have verified that artificially increasing the SFR of all model galaxies (i.e., independent of their positions within the simulation) by this amount does not change our conclusions, as it would imply even smaller correlation lengths all luminosities (as can already be argued from Fig. 13).

The AGN removal performed on our sample does not significantly affect the best fit correlation lengths or slopes. However, two points are worth noting. First, the fraction of AGN candidates is higher among LIRGs (17%) than in the total samples (8%), consistent with what observed for IRAS galaxies in the local Universe, where a higher fraction of AGN is found in more luminous IR objects (e.g., Sanders & Mirabel 1996). Second, a small ($\sim$$5{-}7\%$) systematic decrease of the correlation lengths is observed when AGN are removed from the samples, which is consistent with the fact that AGN in GOODS (which have r₀=5-10 h^-1 Mpc, Gilli et al. 2005) are more strongly clustered than is the full IR galaxy population.

Figure 13: From top to bottom panel: best fit correlation length, slope and amplitude, for the total, $L_{\rm IR}>10^{10}\;L_{\odot}$ and $L_{\rm IR}>10^{11}\;L_{\odot}$ samples obtained by combining the GOODS-S and GOODS-N fields (see Table 2). The best fit clustering parameters are plotted at the sample median $L_{\rm IR}$. The shaded areas show the average and dispersion of the best fit clustering parameters measured over 40 mock fields with the dimensions of a GOODS field (see text for details).

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5.3 Implications for the cosmic variance of 24 $\mu$m source counts

The measured clustering level of star forming galaxies implies that important field-to-field variations should be observed in the number counts of these sources. As discussed in Sect. 2, we have in fact found that the surface densities in GOODS-N versus GOODS-S field differ at the 20% level, once spectroscopic incompleteness is taken into account. Given our direct clustering measurements, we can verify a posteriori if this difference may be understood in terms of cosmic variance in the counts. The expected total variance in the counts can be expressed as:

$$\sigma^2 = N(1+N \times IC)$$ (7)

where N is the average number of galaxies observed and IC is the integral constraint (see, e.g., Daddi et al. 2000, for definitions), which depends on the angular clustering amplitude A and can be related to it following Roche et al. (1999). We have used the best fit clustering parameters r₀ and $\gamma$, Limber's equation, and the observed redshift distribution functions (Fig. 5) to compute that sources with $f_{24}>20~\mu$Jy should have an angular clustering amplitude of $A(1^\circ)\sim0.008$. Given the values of the angular correlation amplitude and slope, and the size of the GOODS fields we infer an integral constraint of 0.13. By inserting these values in Eq. (7) one sees that $N \times IC \gg 1$, i.e., that fluctuations in the number counts of galaxies with $f_{24}>20~\mu$Jy in GOODS-sized fields are dominated by clustering (i.e., cosmic variance) rather than counting (Poisson) statistical uncertainties. We expect fluctuations at the level of 35% ($1\sigma$) in the counts for $f_{24}>20~\mu$Jy galaxies in GOODS-sized fields, fully explaining the observed difference between GOODS-S and GOODS-N.

  
6 Discussion

6.1 Comparison with other galaxy samples at z $\sim$ 1

Deep redshift surveys such as VVDS and DEEP2 are providing an accurate census of the galaxy population at $z\sim 1$, measuring in particular the dependence of galaxy clustering on several parameters such as the galaxy spectroscopic type, color and luminosity. In both surveys, galaxies which can be identified as star forming appear to have a correlation length smaller than that measured for our GOODS 24 $\mu$m selected sample, although the significance of this difference is still limited. In detail, Coil et al. (2004) find $r_0=3.2\pm0.5~h^{-1}$Mpc for emission line galaxies in DEEP2 ($\sim$$1.3\sigma$ lower than that for the total GOODS $24~\mu$m sample), while Meneux et al. (2006) find $r_0=2.5\pm0.4~h^{-1}$ Mpc for star forming, blue galaxies in the VVDS ($\sim$$2.7\sigma$ lower than the total GOODS $24~\mu$m sample). The main difference between the GOODS sample considered here and those from DEEP2 and VVDS resides in the selection at mid-IR versus optical wavelengths. The required detection of sources at 24 $\mu$m for GOODS (in particular the requirement of $f_{24}>20~\mu$Jy) imposes a lower limit to SFR of about 2.5 $M_{\odot}$ yr^-1 at $z\sim 0.8$ (see Fig. 4), while optical selection ( $I_{\rm AB}<24$ mag and $R_{\rm AB}<24.1$ mag for VVDS and DEEP2 galaxies, respectively) does not translate as directly into a SFR. Indeed, because of older stars and dust extinction, even galaxies with very similar optical properties could span a very wide range of star formation rates. We verified that if we impose a cut in SFR or 24 $\mu$m flux density on the Millennium mock catalogs, many low-SFR objects excluded from the sample would be included if a simple optical magnitude cut had been used instead (e.g., $z_{\rm AB}<23.5$ mag, the limit for optical spectroscopy of GOODS sources considered here). In fact, the median SFR of Millennium mock sources increases by a factor of $\sim$6 when the additional mid-IR cut is included. Therefore, in optically selected samples, star forming galaxies are expected to have a lower star formation rate on average than that of our MIPS sources. The trend discussed in the previous section, in which r₀is larger for samples selected at increasing $L_{\rm IR}$ (or SFR), is in line with this interpretation. In connection with the above considerations, it is interesting to note that the strong clustering level measured for GOODS LIRGs appears then to be more similar to that measured for passive galaxies than for moderately star forming galaxies at $z\sim 1$ (Coil et al. 2004, see also Fig. 14). Since the amplitude of galaxy clustering is directly related to the galaxy mass (on average, more massive galaxies reside in denser, i.e., more clustered, environments), this result is in agreement with the observed dichotomy for massive galaxies at $z\la 1.2$, most of which either have already ceased forming stars, or are doing so at very high rates (Noeske et al. 2007; Elbaz et al. 2007).

6.2 Comparison with predictions of galaxy formation models

In Sect. 5 we showed that MIPS detected sources in the GOODS fields appear to be significantly more clustered than expected from galaxy formation models based on the Millennium simulation (Kitzbichler & White 2007). One may wonder if this discrepancy can be ascribed to uncertainties in the SFR to $L_{\rm IR}$ conversion, since $L_{\rm IR}$ is the available (although indirect) measurement for real data, while SFR is the primary output for mock galaxies. Under different assumptions on the stellar IMF the overall uncertainties in the SFR to $L_{\rm IR}$ relation can be quantified to about 30%. We verified that a 30% variation of the 24 $\mu$m flux density threshold in the mock catalog does not alter significantly the Millennium correlation function.

As emphasized by Elbaz et al. (2007), at $z\sim 1$ Millennium galaxies are forming stars at rates about a factor of 3 lower than observed galaxies. As far as object selection is concerned, artificially increasing the SFR of model galaxies is equivalent to selecting galaxies in the mock sample at lower 24 $\mu$m flux densities. This selects many more sources, which are in general less clustered since the lower tail of the SFR distribution is now being sampled. We checked that reducing the limiting f₂₄ flux density by a factor of 3 produces a lower correlation function for Millennium sources, thus reinforcing the discrepancy with the real data. To be fair, it should be noted that simulated galaxies are free from some of the observational selection effects which affect real data in our samples and complicate a direct comparison. For example, at the faintest flux limits of $f_{24}\approx 20~\mu$Jy, where $S/N\sim5$ for MIPS detections, we might be failing to detect sources in crowded regions or close to brighter mid-IR targets. We expect this should be a small effect, but not entirely negligible and in any case difficult to properly simulate. Also, the 50-65% spectroscopic completeness may introduce a bias if sources with measured redshifts have different clustering properties from sources without redshifts (i.e., if sources with redshifts are not a random sampling of the full population). For example, some tendency is detected in both fields for larger spectroscopic completeness at brighter z-band magnitudes (see Fig. 1). Therefore the observed discrepancy between the GOODS data and the mock catalogs from Millennium should be considered by keeping in mind those caveats.

At any rate it is interesting to investigate what could be a likely ingredient that has to be modified within the semi-analytic models in Millennium to explain the observed discrepancy. We suggest here that a possible weakness in the models is the SFR algorithm adopted for the mock galaxies. Indeed, within simulated dense environments like galaxy clusters and groups, a very abrupt cut-off of gas-cooling is applied to galaxies as soon as they become non-central. Therefore, simulated satellite galaxies might be not forming stars at sufficiently high rates, which would indeed reduce the correlation length of the star forming simulated population as well as their number density (see the next section).

Figure 14: Correlation length and space density of GOODS "all'' ( $f_{24}>20~\mu$Jy) and LIRGs galaxy samples considered in this work are compared to that of other galaxy populations at $z\sim 1$, as labeled. The trend predicted from the Millennium simulation for dark matter halos at $z\sim 1$ above different mass thresholds is also shown as a shaded region. More massive halos (log of the threshold mass is labeled) are less abundant and more clustered than less massive ones. GOODS IR galaxies and the absorption line galaxies of Coil et al. (2004) appear more abundant than the halos that can host them (i.e., having the same r0 value), suggesting the presence of more than one galaxy per halo. As discussed in the text, the corresponding IR galaxies and LIRGs at $z\sim 1$ in the mock galaxy catalogs based on Millennium appear significantly less clustered than observed in GOODS. Moreover, Millennium LIRGs are also significantly less abundant than GOODS LIRGs. Values plotted for Millennium LIRGs and IR galaxies were derived averaging measurements in 40 GOODS-sized mock fields.

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6.3 The connection with dark matter halos

While at small scales, comparable to the dimensions of dark matter halos, the clustering of a given galaxy population is difficult to predict because of merging and interactions that can trigger a number of physical processes, at larger scales (e.g., >$1\;h^{-1}$ Mpc), where galaxy interactions are rare, the galaxy correlation function should follow that of the hosting dark matter halos. An interesting consequence is that one can estimate the masses of the typical halos hosting a given galaxy population by simply comparing their clustering level (see, e.g., Giavalisco & Dickinson 2001). Indeed, according to the standard $\Lambda$CDM hierarchical scenario, dark matter halos of different mass cluster differently, with the more massive halos being more clustered for any given epoch, and it is then straightforward to compute the correlation function for halos above a given mass threshold. It is worth noting that since less massive halos are more abundant, the correlation function of halos above a given mass threshold is very similar to the clustering of halos with mass close to that threshold. Also, it is important to note that as far as our clustering measurements are concerned (see Sect. 5), the $w(r_{\rm p})$ datapoints at large scales ( $r_{\rm p}>1\;h^{-1}$ Mpc) have smaller errorbars and guide the power law fit (see Fig. 8). Therefore the measured r₀ and $\gamma$ values are essentially due to the clustering signal at large scales, where the galaxy correlation function follows that of the dark matter, allowing a meaningful comparison with the clustering expected for dark matter halos.

We considered the dark matter halo catalogs available for the milli-Millennium simulation, a reduced version of the Millennium run which includes 1/512 of the full simulated volume. Halo catalogs are available at different time steps along the simulation. Here we considered those at $z\sim 1$ (parameter stepnum=41 in the simulation). In total there are about 32 000 halos with mass above $10^{10}\;M_{\odot}$ in a cubic volume of 62.5$\;h^{-1}$ Mpc on a side. We computed the correlation function and the space density of halos above mass thresholds of log( $M/M_{\odot})=10.8$, 11.2, 11.6, 12.0, 12.4, 12.8. Here we use as halo mass estimator the simulation parameter m_Crit200, defined as the mass within the radius where the integrated halo overdensity is 200 times the critical density of the simulation. The results are shown in Fig. 14, where it is readily evident that more massive halos are more clustered and less numerous. The halo region plotted in Fig. 14 takes into account the fluctuations in the halo space density due to cosmic variance on volumes equal to the milli-Millennium volume (see Sect. 5.3 and Somerville et al. 2004, for a description of the methods to derive the fluctuations in the source counts from the clustering parameters).

We computed the space density of sources in our GOODS samples and compared the r₀ and density values of our populations with those of other galaxy populations at $z\sim 1$ and with those of dark matter halos at $z\sim 1$ as computed above. Comparable values for the space densities of GOODS sources were found when considering the full z=0.1-1.4 redshift range or a restricted redshift interval ( z=0.7-1.2) around the peak of the selection function. The comparison is shown in Fig. 14. Conservative uncertainties of 50% have been considered in the galaxy space densities, which should take into account the fluctuations due to cosmic variance as well as the uncertainties in the volume effectively spanned by the considered galaxy populations. By comparing the halo and the galaxy r₀values, one can immediately see that $f_{24}>20~\mu$Jy star forming galaxies are hosted by halos with masses $\ga$ $8 \times 10^{11}\;M_{\odot}$, while LIRGs, which are more clustered, are on average likely hosted by more massive halos with $M\ga 3 \times 10^{12}\;M_{\odot}$. The population of absorption-line galaxies by Coil et al. (2004) also appears to be hosted by massive halos ( $M\ga 5 \times 10^{12}\;M_{\odot}$), while their emission line galaxies seem to reside in smaller halos with $M\ga 4 \times 10^{11}\;M_{\odot}$. When looking at their space densities, $f_{24}>20~\mu$Jy star forming galaxies (and LIRGs) and absorption line galaxies at $z\sim 1$ appear more abundant than halos that can host them, i.e., there is likely more than one such galaxy per halo. This is consistent with our measurements of $w(r_{\rm p})$. Indeed, as shown in Fig. 8, the clustering signal is well detected down to very small scales of $r_{\rm p}=60\;h^{-1}$ kpc, well within the typical size of dark matter halos. As an example, the average half-mass radius for Millennium halos with $M>8 \times 10^{11}\;M_{\odot}$, i.e., those which likely host GOODS IR galaxies, is about 100 h^-1 kpc. Therefore, most of the signal at scales $r_{\rm p}\la 0.3\;h^{-1}$ Mpc is likely dominated by galaxies within the same halos (i.e., the so-called intra-halo term) and a steepening of $w(r_{\rm p})$ is indeed consistently observed at these scales (Fig. 8). A fully consistent analysis of mid-IR galaxy clustering within the halo occupation number (HOD) theoretical framework (e.g., Peacock & Smith 2000; Moustakas & Somerville 2002; Kravtsov et al. 2004) is however beyond the scope of this paper.

To conclude this section we note that Millennium simulated star forming galaxies and LIRGs at $z\sim 1$ are less clustered than observed in GOODS and that, moreover, observed LIRGs appear significantly more abundant than those in Millennium (Fig. 12). This further supports the interpretation that, at $z\sim 1$, many galaxies within dense environments such as groups or clusters are forming stars at high rates, in contrast to the star formation history assumed in the Millennium simulation. The model's scarcity of star forming galaxies in dense environments, e.g., within the same dark matter halo, may be also responsible for the observed flattening of the Millennium correlation function towards small scales (see Fig. 8).

It is not clear yet what is the main driver of star formation in galaxies at $z\sim 1$. On the one hand, a correlation between star formation rate and galaxy mass is observed (Noeske et al. 2007; Elbaz et al. 2007). On the other hand, as found in this work, higher star formation rates are hosted by galaxies in denser environments. These two results are perfectly consistent one another (and with the conclusions of Elbaz et al. 2007 and Cooper et al. 2007), since more massive galaxies are indeed located in dense environments, but it is hard to establish what is the ultimate driver for the star formation increase: is it the galaxy mass or the environment? In other words, is the star formation rate in each galaxy simply linked to the gas mass and triggered at a given time along the galaxy life almost independently of the environment or, instead, are environmental effects necessary to produce gas instabilities and trigger star formation? Solving these issues is beyond the scope of this paper. It will require much larger samples of star forming galaxies with spectroscopic redshifts, with which one will be able to study clustering of galaxies versus their star formation rates in narrow mass bins.

  
6.4 Descendants and progenitors of z $\sim$ 0.5-1 star forming galaxies

Under simple assumptions, the spatial clustering of an extragalactic source population measured at a given epoch can be used to estimate the typical dark matter halos in which these objects reside, and then to estimate their past and future history by following the halo evolution in the cosmological density field. A useful quantity for such analyses is the bias factor, defined as $b^2(r,z,M)=\xi_{\rm g}(r,z,M)/\xi_{\rm m}(r,z)$, where $\xi_{\rm g}(r,z,M)$ and $\xi_{\rm m}(r,z)$ are the correlation function of the considered galaxy population and that of dark matter, respectively. In general the bias parameter can be a function of scale r, redshift z, and object mass M. For simplicity we adopt the following definition here:

$$b^2(z)=\xi_{\rm g}(8,z)/\xi_{\rm m}(8,z)$$ (8)

in which $\xi_{\rm g}(8,z)$ and $\xi_{\rm m}(8,z)$, are the galaxy and dark matter correlation function evaluated at 8 h^-1 Mpc, respectively. The galaxy correlation function has been measured directly in this work, while the dark matter correlation function can be estimated using the following relation (e.g., Peebles 1980):

$$\xi_{\rm m}(8,z)=\sigma_8^2(z)/J_2$$ (9)

where $J_2=72/[(3-\gamma)(4-\gamma)(6-\gamma)2^\gamma]$ and  $\sigma_8^2(z)$ is the dark matter mass variance in spheres of 8 h^-1 Mpc comoving radius, which evolves as $\sigma_8(z)=\sigma_8(0)D(z)$. D(z) is the linear growth factor of perturbations, while $\sigma_8=\sigma_8(0)$ is the rms dark matter fluctuation at present time, which we fix to $\sigma_8=0.8$ in agreement with the recent results from WMAP3 (Spergel et al. 2007). While in an Einstein-De Sitter cosmology the linear growth of perturbations is simply described by $D_{\rm EdS}(z)=(1+z)^{-1}$, in a $\Lambda$-dominated cosmology the growth of perturbations is slower. We consider here the so-called growth suppression factor $g(z)=D(z)/D_{\rm EdS}(z)$ as approximated analytically by Carroll et al. (1992).

The above relations allow us to estimate the bias of the galaxy population at its median redshift. One can further assume that the spatial distribution of the observed galaxy population simply evolves with time under the gravitational pull of growing dark matter structures. This scenario, in which galaxy merging is considered negligible, is often called the galaxy conserving model and in this case the bias evolution can be approximated by

b(z)=1+[b(0)-1]/D(z) (10)

where b(0) is the population bias at z=0 (Nusser & Davis 1994; Fry 1996; Moscardini et al. 1998).

Once b(z) is determined, the evolution of $\xi_{\rm g}(8,z)$ and hence of r₀(z) can be obtained by inverting Eq. (8). The best fit $\gamma\sim 1.5$ values found in this work are assumed in the above relations. In Fig. 15 we show the evolution of b(z)and r₀(z) for the combined GOODS samples reported in Table 2. Star forming ( $f_{24}>20\;\mu$Jy) objects at $z\sim 0.7$ are expected to have $r_0\sim6{-}7\:h^{-1}$ Mpc at a redshift of 0.1. Since local early type galaxies with L<L_* have been observed to be clustered that strongly in the SDSS and 2dFGRS (Zehavi et al. 2002; Madgwick et al. 2003), at least part of them could descend from $z\sim 0.7$ star forming objects. Similarly, some of the brighter ($L\sim L_*$) ellipticals in the local Universe, for which $r_0\sim8~h^{-1}$ Mpc has been measured (Guzzo et al. 1997; Budavari et al. 2002) could descend from $z\sim 1$ LIRGs ( $L_{\rm IR}>10^{11}~L_{\odot}$), which are expected to evolve into a population with $r_0\sim7{-}8\:h^{-1}$ Mpc by z=0. This would be consistent with the recent findings by Cimatti et al. (2006), who observe a lower number density of $L\la L_*$ early type galaxies at $z\sim 0.8$ than at z=0, suggesting that at least part of local ellipticals have formed since $z\sim 1$.

Figure 15: Bias ( upper panel) and correlation length ( lower panel) for the total, $L_{\rm IR}>10^{10}~L_{\odot}$ and $L_{\rm IR}>10^{11}~L_{\odot}$combined GOODS samples quoted in Table 2, compared with evolutionary tracks computed according to a conserving scenario (solid lines, see text for details). The shaded area shows the r0 evolution of $z\sim 2$ star forming galaxies as computed by Adelberger et al. (2005).

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The slope of the correlation function for local ellipticals is generally found to be steeper than that observed for GOODS IR galaxies. Slopes of $\gamma\sim 1.9{-}2$ have indeed been measured for local ellipticals (Guzzo et al. 1997; Zehavi et al. 2002; Madgwick et al. 2003), as opposed to $\gamma\sim1.5{-}1.6$ for GOODS star forming galaxies measured in this work. While an average steepening of the matter correlation function and of the overall galaxy population is expected towards lower redshifts (see, e.g., Kauffman et al. 1999; Moustakas & Somerville 2002) since the clustering level progressively increases at smaller scales, the clustering evolution in the proposed galaxy conserving scenario above is computed by assuming a fixed ( $\gamma=1.5$) slope. Also it has to be kept in mind that the galaxy conserving scenario is an ideal, rather extreme, representation of galaxy evolution, since it, by definition, neglects galaxy merging. It is therefore somewhat misleading to determine the descendants of a high redshift galaxy population simply based on the r₀ comparison without considering the slope. A $z\sim 1$ star forming galaxy does not evolve automatically into a z=0 elliptical and perhaps subsamples of the local spiral galaxy population may have the clustering properties expected for the descendants of $z\sim 1$star forming galaxies. In an SDSS-based paper, Budavari et al. (2002) have analyzed the clustering properties of $z\sim 0.2$ galaxies with different spectral energy distributions (SEDs) corresponding to those of galaxies with different morphological types. They found that bright ( -23<M_R<-21) galaxies with SEDs corresponding to the morphological type Scd have a correlation length of r₀=6.75 h^-1 Mpc, similar to those of ellipticals at the same redshift, but with a shallower slope $\gamma\sim1.7$. We suggest that part of the GOODS LIRGs population may then evolve into bright, massive spirals. By adding the space densities of local ellipticals and bright spirals one further sees that this is similar to what is measured for $z\sim 1$ star forming galaxies.

Recently, Adelberger et al. (2005) measured the clustering of star forming galaxies at $z\sim1.5{-}2$ (BM and BX samples) and at z=3(LBGs, see also Giavalisco & Dickinson 2001). By comparing the galaxy correlation function with that of dark matter halos in the $\Lambda$CDM-GIF simulation (Kauffmann et al. 1999), Adelberger et al. (2005) found that UV selected galaxies at $z\sim 2$ are hosted by halos with masses around $10^{12}~M_{\odot}$. Furthermore, by following the evolution of these halos in catalogs computed at subsequent time steps in the simulation, they were then able to infer the correlation length of the descendants of the $z\sim 2$ galaxy population. At $z\la 1$ they find that the only galaxy population with clustering strong enough to be consistent with that of the expected descendants of UV selected galaxies are red absorption line dominated galaxies from Coil et al. (2004). In Fig. 15 the expected evolution of $z\sim 2$ starburst galaxies as computed by Adelberger et al. (2005) is also shown. The clustering length of LIRGs at $z\sim 1$is large enough to be consistent with the one predicted for the descendants of UV selected galaxies. Moreover, the correlation slopes of the two populations are similar ( $\gamma\sim1.5{-}1.6$). The average SFR of UV-selected galaxies is also of the same order of that of LIRGs (35 $M_{\odot}$ yr^-1 on average.) It is therefore possible that LIRGs at $z\sim0.5{-}1$, in addition to passive galaxies, may be the direct descendants of UV-selected galaxies. This would imply, in turn, that star formation in these galaxies is sustained, either continuously or intermittently, over cosmological timescales of a few Gyrs and suggests they assemble stellar masses up to $\sim$ $10^{11}\;M_{\odot}$ from $z\sim3$ to $z\sim 1$. Our conclusions on the $z\sim 1$ descendants of high redshift star forming galaxies add to those reached by Adelberger et al. (2005), who, based on the comparison with the correlation lengths measured in the DEEP2 surveys, identify passive absorption line galaxies at $z\sim 1$ as the descendants of their LBG population. DEEP2 star forming objects were on the contrary ruled out based on their small correlation length. As explained in the previous section, the low correlation length of emission line (star forming) galaxies in the DEEP2 survey can be ascribed to a SFR on average lower than that measured for our LIRGs. Our results suggest that star formation is intense in a significant fraction of massive objects at $z\sim 1$ and that the descendants of high redshift star-forming galaxies have not necessarily stopped forming stars at $z\sim0.5{-}1$. If we consider that LIRGs and passive galaxies at $z\sim 1$ have similar space densities ($\sim$ $2.5{-}3\times\;10^{-3}$ Mpc^-3, Fig. 14), and that their combined density is of the order of the LBG space density ($\sim$ $4{-}6\times10^{-3}$ Mpc^-3), then we can conclude that a significant fraction of $z\sim 2$ star forming galaxies might still be forming stars at $z\sim 1$.

7 Summary and conclusions

We present the first measurements of the spatial clustering of star forming galaxies at $z\sim 1$ selected at 24 $\mu$m by Spitzer/MIPS in the GOODS-S and GOODS-N fields. The correlation length for the total combined sample has been found to be $4.0\pm0.4~h^{-1}$ Mpc, the r₀ value in GOODS-S being $\sim$$10\%$larger than in GOODS-N. We estimate the uncertainties in our measurements using mock catalogs extracted from the Millennium simulation, which show that the GOODS-S and GOODS-N measurements are fully consistent with the expected cosmic variance on these 160 arcmin² fields. We find indications for an increase of the correlation length with $L_{\rm IR}$ (or SFR), with LIRGs having $r_0=5.1\pm0.8~h^{-1}$ Mpc. The measured correlation length in the GOODS mid-IR selected samples appears larger than that measured in optical samples of star forming galaxies at $z\sim 1$ such as those in the DEEP2 or the VVDS surveys. Although the significance of this result is still limited ( $1{-}3\sigma$), it might be interpreted as evidence that the average star formation rate in optically selected samples of emission line galaxies is lower than that of our samples, which, by selection, have larger IR luminosity. This is in agreement with the observed relation between IR luminosity and clustering strength, which, in turn, suggests that at $z\sim 1$ more intense star formation is hosted by more massive (i.e., more clustered) systems.

The measured correlation length is significantly larger than that expected from the Millennium simulations, once the selection criteria adopted to define the real data samples are applied to the mock samples. This suggests that star formation is, on average, occurring in dark matter halos that are more massive than those predicted by the galaxy formation model implemented in the Millennium simulation by Croton et al. (2006). By comparing the clustering of GOODS star forming galaxies with that of Millennium dark matter halos, we find that more luminous galaxies are hosted by progressively more massive halos, with LIRGs residing in halos with $M\ga 3 \times 10^{12}~M_{\odot}$. Since the measured LIRG space density is higher than that of the hosting halos, each halo appears to contain on average more than one LIRG. This is also supported by the steepening of the correlation function observed towards smaller scales, which is usually interpreted as due to galaxy pairs within the same dark matter halo (intra halo clustering).

Based on a galaxy conserving scenario, in which it is assumed that galaxies observed at a given redshift evolve without merging, simply pulled by the surrounding density field, we trace the time evolution of the bias parameter and of the correlation length of $z\sim 1$star forming galaxies. By comparing the evolved correlation lengths with those of local and high-redshift galaxy samples, we infer the likely descendant and progenitors of our $z\sim 1$ sample. We find that objects in our sample may evolve into L<L_* ellipticals or bright spirals by z=0, with LIRGs evolving into bright $L\sim L_*$objects. Similarly, LIRGs, together with passive absorption line galaxies at $z\sim 1$, may be identified as the descendants of UV-selected star forming galaxies at $z\sim 2$.

Acknowledgements

We wish to thank the referee for comments which improved the paper significantly. We acknowledge G. Zamorani, L. Pozzetti, F. Pozzi, C. Gruppioni, L. Moscardini, E. Branchini and M. Magliocchetti for useful discussions. We are also grateful to E. MacDonald and H. Spinrad for their extensive efforts obtaining some of the redshift measurements used in this work. R.G. acknowledges financial support from the Italian Space Agency (under the contract ASI-INAF I/023/05/0) and from the grant PRIN-MUR 2006-02-5203. The work of D.S. was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.

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