In hierarchical cosmologies the evolution of galaxy clustering depends both on cosmological quantities such as Ω, Λ and P(k), which determine how collapsed structures — dark matter haloes — form and evolve, and on the physical processes — cooling, star formation, radiative and hydrodynamic feedback — which drive the formation of galaxies within these merging haloes. In this paper we combine dissipationless cosmological N-body simulations and semi-analytic models of galaxy formation in order to study how these two aspects interact. We focus on the differences in clustering predicted for galaxies of differing luminosity, colour, morphology and star formation rate, and on what these differences can teach us about the galaxy formation process. We show that a ‘dip’ in the amplitude of galaxy correlations between z=0 and z=1 can be an important diagnostic. Such a dip occurs in low-density CDM models, because structure forms early, and dark matter haloes of mass ∼1012 M⊙, containing galaxies with luminosities ∼L*, are unbiased tracers of the dark matter over this redshift range; their clustering amplitude then evolves similarly to that of the dark matter. At higher redshifts, bright galaxies become strongly biased and the clustering amplitude increases again. In high density models, structure forms late, and bias evolves much more rapidly. As a result, the clustering amplitude of L* galaxies remains constant from z=0 to z=1. The strength of these effects is sensitive to sample selection. The dip becomes weaker for galaxies with lower star formation rates, redder colours, higher luminosities and earlier morphological types. We explain why this is the case, and how it is related to the variation with redshift of the abundance and environment of the observed galaxies. We also show that the relative peculiar velocities of galaxies are biased low in our models, but that this effect is never very strong. Studies of clustering evolution as a function of galaxy properties should place strong constraints on models of galaxy formation and evolution.
Local galaxies are highly clustered. On large scales, they are organized into a network of sheets and filaments which surround large underdense regions, usually referred to as voids. On smaller scales, they are found in gravitationally bound groups and clusters. According to the standard theoretical paradigm, the structures observed today were formed by the gravitational amplification of small perturbations in an initially Gaussian dark matter density field. Small-scale overdensities were the first to collapse, and the resulting objects subsequently merged under the influence of gravity to form larger structures such as clusters and superclusters. Galaxies formed within dense haloes of dark matter, where gas was able to reach high enough overdensities to cool, condense and form stars.
In this hierarchical formation picture, the clustering of the dark matter, as measured by the amplitude of the matter correlation function ξm(r), increases monotonically with time. The precise evolution of ξm(r) with redshift has been studied extensively using both N-body simulations (e.g. Jenkins et al.1998) and analytic methods (Hamilton et al. 1991; Peacock & Dodds 1994; Jain, Mo & White 1995). If ξm(r, z) were observable, it would be straightforward to use its behaviour to determine Ω, Λ and the power spectrum of linear density fluctuations. What one measures in practice, however, is the clustering of galaxies, and the interpretation then requires an understanding of how these objects trace the underlying dark matter density field.
If galaxies form at the centre of dark matter haloes, considerable insight may be gained by using N-body simulations to study the clustering of haloes (Brainerd & Villumsen 1994; Mo & White 1996; Roukema et al. 1997; Jing & Suto 1998; Bagla 1998a,b; Ma 1999; Wechsler et al. 1998). Mo & White (1996) tested an approximate analytic theory against their numerical results and this theory and its extensions can also be used to analyse the evolution of halo clustering with redshift (Matarrese et al. 1997; Coles et al. 1998). An important conclusion from all these studies is that the clustering of haloes of galactic mass (∼1012 M⊙) evolves much more slowly than the clustering of the dark matter. This is because at high redshifts, such haloes correspond to rare peaks in the initial density field, and are thus more strongly clustered than the dark matter (Kaiser 1984). Another important conclusion is that more massive haloes are more strongly clustered than less massive haloes. If the luminosity of a galaxy is correlated with the mass of its halo, more luminous galaxies ought to be more strongly clustered. A detailed comparison with observational data requires a model for the observable properties of the galaxies present within haloes of given mass at each epoch.
Precise measurement of the clustering amplitude of galaxies at high redshift has just recently become feasible. Usually this is done by calculating the angular two-point correlation function w(θ) as a function of apparent magnitude. In order to assess how clustering has evolved, w(θ) must be deprojected using Limber's equation under the assumption of some specific model for the redshift distribution of the observed galaxies. In future, large surveys of faint galaxies with photometric and/or spectroscopic redshifts will be available (see, e.g., Connolly et al. 1995). It will be possible to classify the galaxies in these surveys according to absolute magnitude, spectral type, star formation rate and colour, and to investigate how clustering evolution depends on these properties.
At present, most of the data indicate that the clustering amplitude of galaxies decreases from z=0 to z=1. Different analyses, however, yield very different estimates for the strength of this decrease. Le Fèvre et al. (1996) analysed the clustering of 591 galaxies with I<22.5 in the five 10-arcmin fields of the CFRS survey. They find that clustering has evolved dramatically, quoting a comoving correlation length at redshift 0.5 of r0=2 h−1 Mpc (q0=0.5). More recently, Carlberg et al. (1999) presented a preliminary analysis of clustering in the CNOC2 field galaxy redshift survey. Their sample is spread over four patches of sky with a total area of 1.5 deg2. They estimate that the comoving correlation length of galaxies with MR<−20 evolves as r0(z)=(5.15±0.15)(1+z)−0.3±0.2h−1 Mpc. This is much weaker than the evolution found for the CFRS galaxies. The results of Carlberg et al. (1999) agree reasonably well with those of Postman et al. (1999), based on 710 000 galaxies with IAB<24 from an imaging survey of a contiguous 4×4 deg2 region of the sky. The latter authors suggest that the small volumes sampled by the CFRS and other early surveys resulted in their derived correlation lengths being biased low.
The clustering of Lyman-break galaxies at z∼3 has now been measured with surprising precision (Giavalisco et al. 1998; Steidel et al. 1998; Adelberger et al. 1999). The comoving correlation length of these objects is comparable to that of L* galaxies today, implying, as expected from models of halo clustering, that Lyman-break galaxies are highly biased tracers of the dark matter distribution at these redshifts. In addition, Giavalisco et al. (1998) find that the fainter Lyman-break galaxies are less strongly clustered. This accords well with a simple model in which the star formation rates in these objects increase with the mass of their haloes. More detailed theoretical modelling of the observed properties of Lyman-break galaxies at z=3, including analysis of their abundances, sizes, luminosities, colours, star formation rates and clustering properties, has been carried out by Mo & Fukugita (1996), Baugh et al. (1998), Governato et al. (1998), Mo, Mao & White (1999) and Somerville, Primack & Faber (1999).
In this paper we combine cosmological N-body simulations and semi-analytic modelling of galaxy formation to study the evolution of galaxy clustering as a function of redshift. Our methods for incorporating galaxy formation in the simulations are discussed in detail in Kauffmann et al. (1999, hereafter Paper I). Two variants of a cold dark matter (CDM) cosmology are analysed here: a high-density model with Ω=1, Γ=0.2 and H0=50 kms−1 Mpc−1 (τCDM), and a low-density flat model with Ω=0.3, Λ=0.7 and H0=70 kms−1 Mpc−1 (ΛCDM). Paper I was concerned with the global properties of the galaxy distribution at z=0, including B- and K-band luminosity functions, the I-band Tully-Fisher relation, galaxy two-point correlation functions, colour distributions, star formation rate functions and peculiar velocity distributions. Here we focus on clustering evolution in the two models. We study the predicted differences in clustering evolution for galaxies of different magnitude, type and star formation rate, and we outline how future observational data will clarify the galaxy formation process.
What can be learned from the evolution of halo clustering?
Because galaxy formation is complex and involves many poorly understood physical processes, e.g., star formation and radiative and hydrodynamical feedback, it is worthwhile to ask whether the clustering of dark matter haloes can be used to constrain cosmological parameters directly.
In Fig. 1 we plot the correlation length r0 as a function of redshift for haloes of different mass in our two simulations. Here, as in the rest of the paper, all length-scales are expressed in comoving units. Since the correlation functions in the models are not exact power laws, we define r0 as the separation where ξ(r)=1. The smallest haloes resolved in the simulations contain 10 particles and have virial masses ∼2×1011 M⊙. For these objects, r0 initially decreases with redshift, reaches a minimum, and then increases again. The redshift of this minimum is different for the two cosmologies: z∼0.7 for τCDM and z∼1.5 for ΛCDM. Massive haloes do not exhibit the same ‘dip’ in correlation length; their r0 remains constant for a while, then increases at high redshift. Once again, the redshift at which the evolution becomes strong is lower for τCDM than for ΛCDM. This is simply because structure formation occurs later in the τCDM model.
With 10-m telescopes it is now possible to measure the rotation curves of disc galaxies at redshift ∼1 (Vogt et al. 1996). It will be many years, however, before such samples are both large enough and complete enough for an analysis of the clustering evolution of galaxies as a function of their halo mass. In all likelihood, we will have to deal with flux-limited surveys of galaxies for some time to come.
Let us now make the simplifying assumption that each simulated dark matter halo contains one observable galaxy, and that the luminosity of the galaxy increases with the mass of its halo. The correlation function of a flux-limited sample of galaxies of known abundance at redshift z may then be calculated by evaluating ξ(r) for the mass-limited set of simulated haloes which has the same abundance. This is illustrated in Fig. 2, where we plot the correlation length of haloes versus their number density (in units of h3 Mpc−3) at a series of redshifts. As expected, r0 decreases as the number density increases, because the correlation signal becomes dominated by low-mass haloes, which are more weakly clustered. Note that the differences between the τCDM and the ΛCDM models are small at all redshifts. Mo et al. (1999) show a similar plot for haloes at z=3 for four different CDM cosmologies, and find that they all give similar results. There thus appears to be a ‘cosmic conspiracy’ that makes it impossible to infer information about cosmological parameters from the clustering of haloes of given abundance. On the other hand, the uniformity seen in Fig. 2 can be used as a test of the entire class of hierarchical models and of the hypothesis that there is a one-to-one correspondence between halo mass and galaxy luminosity. As discussed by Mo et al. (1999) and by Steidel et al. (1999), this hypothesis works well for the Lyman-break population at z∼3. At low redshifts, the assumption of a one-to-one correspondence between haloes and galaxies must break down, because the abundance of high-mass haloes is larger, and more and more bright galaxies are grouped together in each such halo. The values of r0 plotted in Fig. 2 are then lower limits on the true values.
The evolution of galaxy clustering
In this section we study the evolution of galaxy clustering in the ΛCDM and τCDM simulations. We use the star formation and feedback recipes that resulted in the best fits to the observational data at z=0. As discussed in Paper I, extremely efficient feedback was required in the τCDM model in order to obtain a reasonable fit to the correlation function on scales below 1 h−1 Mpc, and to avoid producing too many galaxies with luminosities below L*. Even so, the model failed to fit the observed bright end of the luminosity function, even when dust extinction was included, and the clustering amplitude was too low on large scales. The ΛCDM model with relatively inefficient feedback resulted in a better overall fit to most of the data at z=0. In Fig. 3 we show the best fits we were able to obtain to the observed B-band luminosity function for the two models. Recall that our models are normalized to match the I-band Tully-Fisher relation of Giovanelli et al. (1997) at a circular velocity of 220 kms−1, and not to match the B-band luminosity function itself. This results in a somehwat poorer fit to the observations than obtained by other groups (see, e.g., Benson et al. 1999).
For simplicity, we do not consider dust extinction in the analysis of this paper, because it is very uncertain how the empirical recipes we adopted the Paper I should be extended to high redshift. This neglect has little effect on our ΛCDM model, but it means that our τCDM model now substantially underpredicts galaxy clustering at z=0. We concentrate below on the relative evolution of r0 rather than on its absolute value, so this problem does not strongly affect our conclusions. The effect of dust extinction is to increase the contribution of cluster galaxies and to decrease the contribution of star-forming field galaxies to statistics such as the correlation function. In the τCDM model this means that galaxies are more biased relative to the dark matter because they reside in larger haloes on average. The bias of large haloes also evolves more rapidly with redshift, and so, as will become clear below, the inclusion of dust will simply enhance the predicted differences in clustering evolution between the τCDM and ΛCDM models.
In Fig. 4 results are shown for galaxies with rest frame B-band magnitudes brighter than −19+5 log h in the ΛCDM simulation. At redshift zero, this corresponds to selecting galaxies brighter than ∼L*. The first three panels in the plot show the evolution of ξ(r) evaluated at r=2,3 and 8 h−1 Mpc (comoving units). The fourth panel shows the evolution of the comoving correlation length r0. For comparison, the dotted line in each panel shows the evolution of the corresponding quantity for the dark matter. Results for τCDM are given in Fig. 5. In each case the redshift extends to the point at which the abundance of L* galaxies becomes too low for reliable estimation of the correlation function (i.e., less than several hundred such objects in our simulation volume).
In the ΛCDM model, the clustering amplitude decreases from z=0 to z=1.5, remains approximately constant from z=1.5 to z=2.5, and then increases again at higher redshift. The dip in clustering amplitude is stronger on small scales: ξ(r) decreases by a factor of 3 at 2 h−1 Mpc, and by a factor 1.5 at 8 h−1 Mpc. The correlation length r0 decreases from 5.5 h−1 Mpc at z=0 to 3.9 h−1 Mpc at z=1.5. This agrees remarkably well with the parametrization of r0 as a function of z quoted by Carlberg et al. (1999). In the τCDM model, the clustering amplitude remains fixed from z=0 to z=1 and then rises steeply at higher redshifts. Note, as mentioned above, that the correlation length at z=0 is low (∼3 h−1 Mpc) in this model.
In Fig. 6 we plot the evolution of the bias b, defined as the square root of the ratio between the galaxy and the dark matter correlation functions:
The four lines on the plot show the bias as a function of redshift evaluated at r=2, 3, 5 and 8 h−1 Mpc. The bias does not depend on r in either model, except at very high redshifts where b is somewhat larger on small scales. The bias evolves much more rapidly in the τCDM model than in the ΛCDM model. Galaxies in the ΛCDM model are unbiased tracers of the mass out to z∼1. Galaxies in the τCDM model have bias values of about 2 at this redshift.
Dependence on luminosity, star formation rate and morphological type
In Figs 7 and 8 we demonstrate that the clustering evolution depends on the way in which galaxies are selected in the simulations. The first panel compares the clustering evolution of galaxies selected in the rest frame B band with that of galaxies selected in the rest frame I band. The second panel compares the clustering evolution of galaxies with M(B)<−19+5 log h with that of galaxies 1.5 mag brighter. The third panel shows what happens if galaxies are selected by star formation rate rather than by luminosity. The fourth panel shows the clustering of early-type galaxies with stellar mass greater than 3×1010 M⊙ [recall from Paper I that these objects form by mergers of two galaxies of similar mass and have M(B)bulge-M(B)total<1 mag].
In the ΛCDM model, we find that the strength of the ‘dip’ in clustering between z=0 and z=1.5 is sensitive to sample selection. The dip is weaker for more luminous galaxies and for galaxies selected in the I band, but stronger for galaxies selected by star formation rate. The clustering amplitude of early-type galaxies is stronger than that of the population as a whole and evolves very little with redshift. In the τCDM model, clustering evolution is less sensitive to sample selection. The clustering always remains fixed out to z∼1 and then rises steeply at higher redshifts. Early-type galaxies are also very strongly clustered in this model, particularly at high redshifts.
What can be learned from these dependences?
We now explain why the evolution of clustering depends on sample selection, and what can be learned about galaxy formation by studying the observed evolution as a function of morphological type, luminosity, colour and star formation rate.
For a given cosmology, the clustering amplitude predicted for a sample of galaxies depends on the masses of the dark matter haloes they inhabit. The evolution of clustering depends on how the mass distribution of these haloes changes with redshift. Additional relevant and observationally accessible information comes from the variation with redshift of the abundance of galaxies in the sample.
As a first example, let us suppose that galaxies with fixed star formation rate are found in smaller haloes at high redshift than at the present time. We would expect the clustering amplitude of a SFR-selected sample to show a stronger dip than a sample of galaxies that tracked haloes of the same mass at all redshifts. We would also expect the abundance of galaxies in a SFR-selected sample to increase more strongly with redshift, because there are many more small haloes than large ones.
As a second example, let us suppose that early-type galaxies are found primarily in massive haloes at all redshifts. As seen in Fig. 1, these galaxies should not exhibit any dip in clustering, and their abundances should decrease strongly at high redshifts because massive haloes are rare objects at early times.
These points are illustrated in detail in Fig. 9, where we plot the evolution of the median halo mass and the comoving number density of galaxies in samples selected in different ways from the ΛCDM simulation. The top three panels show results for galaxies selected according to rest frame B magnitude, rest frame I magnitude and star formation rate. At z=0, all three galaxy samples have the same abundance and occur in haloes of roughly the same mass. Galaxies selected according to star formation rate move to smaller haloes at higher redshift. This effect is simply a result of the parametrization of star formation in our models. Following Kennicutt (1998), we have adopted a star formation law of the form M*aMcoldtdyn, where Mcold is the mass of cold gas in the galaxy, and tdyn is the dynamical time of the galaxy. Since tdyn decreases at higher redshifts, the star formation rates are higher in haloes of the same cold gas content. Galaxies selected in the B band exhibit a weaker trend towards low-mass haloes. In the case of the I-band selection, galaxies trace haloes of roughly the same mass at all redshifts below 2. This is because the I-band magnitude of a galaxy is a measure of its total stellar mass, rather than its instantaneous star formation rate. We thus conclude that SFR-selected samples show the strongest dip in clustering in Fig. 7, because this selection procedure favours galaxies in lower mass haloes at high redshift. Note that galaxies in the SFR-selected samples also exhibit the strongest increase in abundance from z=0 to z=1.5.
The bottom two panels in Fig. 9 show that very bright galaxies and early-type galaxies in the simulation are found in haloes with masses ∼1013 M⊙. As seen in Fig. 1, the clustering of these objects evolves very little from z=0 to z=1.
In Fig. 10 we show the evolution of galaxy abundances and median halo masses for samples selected in the same way from the τCDM simulation. The results are qualitatively similar to those found for ΛCDM. L* galaxies occur in haloes of roughly the same masses (∼1012 M⊙) in both models. The reason why no dip is seen in the τCDM model is because haloes of these masses are more strongly biased at z=1 than in the ΛCDM model, and the decrease in halo mass with redshift for the SFR-selected sample is less pronounced. Note also that the redshift at which the abundance curves peak is higher for ΛCDM than for τCDM. In the ΛCDM simulation, the abundance of early-type galaxies decreases substantially only at redshifts greater than 1.5, whereas in the τCDM simulation, the abundance of ellipticals has already declined by a factor of 3 by z=1.
Evolution of the slope of ?(r)
Fig. 11 shows the evolution of the slope γ of the two-point correlation function for galaxies with rest frame B magnitudes brighter than −19+5 log h in the ΛCDM and τCDM models. We have fitted a power law to ξ(r) over three different ranges in scale: r=1–5 h−1 Mpc, r=5–10 h−1 Mpc and r=1–10 h−1 Mpc.
In the ΛCDM model, the evolution of the slope is stronger on small scales. Over the range 1–5 h−1 Mpc, γ evolves from –2 at z=0 to –1.5 at z=1. On large scales, γ remains approximately constant. Over the range 1–10 h−1 Mpc, γ evolves from −1.85 at z=0 to −1.6 at z=1, and then remains constant. These results appear to be in qualitative agreement with the observations (Postman et al. 1999). These authors find no dependence of γ on magnitude for the bright (I<21) galaxies in their survey. At fainter magnitudes, γ flattens, reaching a value of −1.6 at I=22.5. They also find that the flattening is stronger on smaller angular scales. Neuschaeffer & Windhorst (1995) find similar results from an independent survey carried out at a different wavelength. In the τCDM model, there is very little change in the slope with redshift on any scale.
Evolution of pairwise peculiar velocities
Estimates of dynamical quantities such as Ω or cluster M/L ratios from galaxy data require knowledge not only of the spatial bias in the galaxy distribution, but also of any possible bias in the kinematics of the galaxies relative to those of the dark matter. In Paper I we explored this ‘velocity bias’ for our z=0 models using pairwise velocity statistics, and in Paper III (Diaferio et al., 1999) we do the same using group and cluster velocity dispersions. Here we briefly explore the predicted evolution of velocity bias using pairwise statistics.
The term ‘velocity bias’ has become rather confused in the literature. When it was first introduced by Carlberg, Couchman & Thomas (1990), it referred to processes which would cause a galaxy to have a different velocity to that of the neighbouring dark matter. This gives rise to differing pairwise velocities for galaxies and for dark matter even when there is no spatial bias. We are not able to study effects of this kind in these simulations. However, spatial bias will affect the pairwise velocity statistics, because velocity dispersion is correlated with environment. If galaxies occur preferentially in dense environments, one would measure a higher mean velocity dispersion for a sample of galaxies than for a random sample of dark matter particles.
The thin solid lines in Fig. 12 show the redshift evolution of the pairwise peculiar velocity dispersion σ12 evaluated at relative separations r=0.5, 1 and 2 h−1 Mpc (comoving units) for galaxies with rest frame B-band magnitudes less than −19+5 log h in the ΛCDM and τCDM simulations. The thick solid lines show the evolution of σ12 for the dark matter. In order to compare the relative change in σ12 as a function of redshift in the two models, we scale the results by dividing by the value of at z=0. As shown in fig. 13 of Paper I, (r=1 h−1 Mpc) in both the ΛCDM and τCDM models at the present time.
The pairwise peculiar velocities of the galaxies follow those of the dark matter quite closely in both models. The galaxy velocities are 10–40 per cent lower than those of the dark matter at z=0. (Note that the ‘antibias’ in galaxy peculiar velocities at z=0 is stronger than that shown in Fig. 13 of Paper I, because the models presented in this paper do not include dust extinction. Dust reduces the contribution of star-forming field galaxies in a B-selected sample, but has little effect on early-type galaxies in rich groups and clusters. Models that include dust extinction thus give values of that are 10–25 per cent larger). The difference between the galaxy and dark matter peculiar velocities decreases at higher redshift. In contrast to the spatial distributions, galaxy peculiar velocities in our models are never very strongly biased. In the τCDM model, there is nearly a factor of 2 decrease in from z=0 to z=0.5. In the ΛCDM models, remains roughly constant out to z=0.5, before decreasing at higher redshift.
Discussion and conclusions
In a hierarchical universe, the evolution of galaxy clustering depends on the following factors.
Cosmological parameters, such as Ω, Λ, σ8 and the shape of the power spectrum, because these determine the rate at which structure grows and the epoch at which haloes of given mass change from being rare objects, and thus biased tracers of the dark matter distribution, to being ‘typical’ objects with clustering properties similar to that of the mass.
The relationship between the mass of a dark matter halo and the properties of the galaxies that form within it. Note that this relationship depends only on halo mass and is independent of the environment in which the halo finds itself (Lemson & Kauffmann 1999).
The evolution of the galaxy population with redshift (and hence the evolution of the relationship between galaxy properties and halo mass).
In this paper we illustrate how differences in clustering evolution between galaxies of differing luminosity, colour, morphological type and star formation rate may help constrain galaxy formation models, and perhaps even cosmological parameters.
One interesting diagnostic that we highlight is the ‘dip’ in correlation amplitude observed between z=0 and z=1. We show that this dip occurs naturally in a ΛCDM model, where structure forms early and haloes with masses in the range 1011–1012 M⊙, which contain galaxies of intermediate luminosities, are unbiased tracers of the mass over this redshift range. In the τCDM model, bias evolves rapidly, and the clustering amplitude of L* galaxies remains constant from z=0 to z=1. Although it might be possible to ‘force’ a dip in clustering in the τCDM model by requiring that L* galaxies form in less massive haloes, it would be difficult to come up with a physically motivated scheme for doing this that would not simultaneously produce too many bright galaxies.
We also show that the strength of the dip in the ΛCDM model is sensitive to sample selection. If galaxies are selected according to star formation rate rather than B-band luminosity, objects in low-mass haloes contribute more to the clustering signal at high redshifts and the dip is stronger. If galaxies are selected in the red rather than the blue, the dip is reduced. Very luminous galaxies and massive early-type galaxies exhibit no dip in clustering between z=0 and z=1, because they occur in high mass haloes (1013–1014 M⊙) that are already biased at z=0 and become substantially more biased at high redshift.
The predictions presented in this paper should be viewed as illustrative rather than quantitative. As discussed in Paper I, the precise relation between the mass of a halo and the properties of the galaxies that form within it depends strongly on the adopted recipes for star formation and feedback; these are very uncertain. The exciting prospect is that future observations of galaxy clustering at high redshift will place strong empirical constraints on these processes.
The simulations in this paper were carried out at the Computer Center of the Max-Planck Society in Garching and at the EPPC in Edinburgh. Codes were kindly made available by the Virgo Consortium. We especially thank Adrian Jenkins and Frazer Pearce for help in carrying them out. We are also grateful to John Peacock for useful discussions. AD is a Marie Curie Fellow and holds grant ERBFMBICT-960695 of the Training and Mobility of Researchers programme financed by the EC.