6 Discussion and comparison with model predictions

   
6.1 Modelling A$_\omega$

For r<20 h^-1 Mpc, the spatial correlation function $\xi(r)$ can be approximated by $\xi(r)=(r_0/r)^{\gamma}$, where, from the results of local redshift surveys, $\gamma \sim -1.8$ and $r_0\sim 4.3 ~{h}^{-1}$ Mpc (Groth & Peebles 1977; Davis & Peebles 1983; Maddox et al. 1990). One way to produce a prediction for the variation of $A_\omega$ with sample limiting magnitude is to assume a functional form for the growth of clustering $\xi(r,z)$, normally written as

$$\xi(r,z)= h(z)\left({r_0\over r}\right)^\gamma ,$$ (6)

where

$$h(z)=(1+z)^{-(3+\epsilon)}.$$ (7)

This functional form is then integrated over redshift space using the relativistic version of Limber's equation, (Phillipps et al. 1978; Groth & Peebles 1977; Limber 1953), assuming that for $\theta \ll 1$ (Efstathiou et al. 1991),

$$\omega(\theta)= \sqrt \pi {\Gamma [(\gamma - 1) /2 ] \over {\Gamma(\gamma/2)}} {A \over {\theta ^{\gamma - 1} }} r_0^\gamma,$$ (8)

where $\Gamma$ is the incomplete gamma function, $\theta$ is the angular separation and A is given by

$$A = \int^\infty_0 g(z)\left( {\rm d}N \over {{\rm d}z} \right... ...\left( {{\rm d}N \over {{\rm d}z}} \right) {\rm d}z \right]^2,$$ (9)

with

$$g(z)={h(z)\over{d_A^{\gamma-1}(z)}}\left ( {{\rm d}r(z)\over{{\rm d}z}}\right )^{-1},$$ (10)

where d_A(z) is the angular diameter distance and ${\rm d}r(z)/{\rm d}z$ is the derivative of the proper distance. In this simple scenario, three cases are of interest: clustering fixed in proper coordinates, in which $\epsilon=0.0$; clustering fixed in co-moving coordinates which gives $\epsilon=-1.2$. Finally, the predictions of linear theory give $\epsilon=0.8$.

Many papers have investigated the scaling of $A_\omega$ with magnitude using the approach detailed above (see, for example, Efstathiou et al. 1991). To interpret measurements of $A_\omega$, using these models, however, involves making at least two critical assumptions: firstly, the form of the redshift distribution  ${\rm d}N/{\rm d}z$ for the faint galaxy population and its evolution as a function of limiting magnitude; and secondly how $\xi(r,z)$ scales with redshift (Eq. (7)). In the following section we examine these two assumptions in turn. (Predicted correlation amplitudes using this formalism are also sensitive to the underlying cosmology, as the size of the volume element at a given redshift is much lower for an Einstein-deSitter cosmology than for a low-$\Omega$ universe. However, to the median redshift of our survey the difference in model predictions between open and flat-Lambda cosmologies small. In this paper we assume that $\Omega_{\rm M}\sim0.1$, in agreement with recent observational evidence.)

   
6.2 Validity of model assumptions

Our ${\rm d}N/{\rm d}z$ is derived from luminosity evolution models which are described fully in Metcalfe et al. (2000). Starting with the observed local galaxy luminosity function and assuming a star-formation history for each galaxy type these models are able to reproduce the observed numbers counts, colours and redshift distributions of the faint galaxy population to the limits of the current observations (Metcalfe et al. 2000). However, at $I_{AB}\sim24$ we may now directly test these model redshift predictions against spectroscopic measurements made in the Hubble Deep Field (HDF) (Cohen et al. 2000; Williams et al. 1996). In Fig. 19 (upper panel) we show the spectroscopic redshift distribution for 120 galaxies in the HDF-North (with 16 non-detections represented as the open box) compared to the predictions of our $\Omega_{\rm M}=0.1,\Omega_\Lambda=0.0$ PLE model distribution (solid line). In the lower panel we show the relationship between median redshift  $z_{\rm med}$ and I_AB limiting magnitude as predicted by this model (note that we compute our median redshift in each case by considering all galaxies brighter than the abscissa magnitude). Additionally, we show measurements from several other I_AB-magnitude limited redshift surveys, including the HDF sample used in the top panel. We also show the median redshift derived from photometric redshifts for two samples limited at I_AB<25 for the HDF N/S, kindly supplied to us by S. Arnouts. The Poissonian error bars computed from all surveys are smaller than the symbols and are not plotted. The true field-to-field variance may of course be much larger, but the fact that we measure the same median redshift for both HDF-N/S catalogues suggest that it is not.

Despite these qualifying remarks, we conclude that our luminosity evolution models provide an acceptable fit to the observed redshift distributions at least to the depth to which we measure galaxy clustering in the CFDF photometric catalogues ( $I_{AB-{\rm med}}\sim24$, or equivalently $I_{AB}\sim24.5$). They are able to reproduce both the trend of  $z_{\rm med}$ with I_AB and the dispersion in redshift at a given magnitude slice. Furthermore, as demonstrated in Metcalfe et al., they also correctly predict the numbers of 2<z<3 galaxies. We therefore conclude that our modelling of ${\rm d}N/{\rm d}z$ is not a major source of uncertainty in our prediction of $A_\omega$.

Figure 19: Upper panel: the redshift distribution of faint galaxies in the Hubble Deep Field North from the compilation of Cohen et al. (2000) (histogram), compared to the predictions of our model (smooth solid line). The open box represents the number of galaxies observed for which no redshift could be determined. Lower panel: the relationship between median redshift  $z_{\rm med}$ predicted by our model (solid line) compared to the CFRS (five-pointed stars), spectroscopic redshifts in the HDF-N (cross) and photometric redshifts from the HDF N/S (asterisk). Poissonian error bars are smaller than the symbols in all cases and are not plotted.

Our second assumption, that the growth of galaxy clustering can be expressed as in Eq. (7), is more problematic. Clustering measurements of Lyman-break galaxies (Adelberger et al. 1998; Giavalisco et al. 1998), have already indicated that the "epsilon'' formalism does not provide an acceptable fit to the observations. Similar results have also been found for measurements of r₀(z) in the HDF-North (Arnouts et al. 1999a). Can our clustering measurements in the CFDF be successfully matched by this model? In Fig. 20 we show our measurements of $A_\omega(1\hbox{$^\prime$ }$) compared to prediction of our models for $\epsilon =-1.2,0.0,1.0,2.0$ (long dashed, solid, dotted and dashed lines respectively), assuming r₀=4.3 h^-1 Mpc, $\rm\Omega_M=0.1, \Omega_\Lambda=0.0$ and $\delta =-0.8$. (We note that clustering predictions for an $\Omega-\Lambda$ cosmology are very similar to zero-$\Lambda$ cosmology). For clarity we omit the literature compilation shown previously. In the magnitude range $18.5<I_{AB-{\rm med}}<22$, we see that our observations are consistent with $\epsilon\sim0$. However, faintwards of $I_{AB} \sim 22$, our observed clustering amplitudes decline more rapidly than the model predictions. By 23.0<I_AB< 24.0 our observations are consistent with $\epsilon\gtrsim 1$. From Fig. 20 it is clear that the $\epsilon\sim0$ model cannot match simultaneously both bright and faint observations in the range $18.5<I_{AB-{\rm med}}<24.0$. Furthermore, rapid growth of clustering for the entire sample ( $\epsilon\sim2$) is marginally excluded because it produces correlations which are already too low by $I_{AB-{\rm med}}\sim 22$ to match our observations. Furthermore, allowing r₀ (i.e., r₀(z=0)) to vary merely changes the normalisation at $I_{AB}\sim18.5$ (which is already in agreement with our observations) but not the slope of the $A_\omega(1\hbox{$^\prime$ })-I_{AB-{\rm med}}$ relation. In Sect. 6.4 we investigate the reason for this discrepancy in more detail.

   
6.3 Colour selected galaxy clustering

In Fig. 18 we clearly see the dependence of $\log(A_\omega)$ on (V-I)_AB colour. To interpret this result, we first note that the dependence of (V-I)_AB colour on redshift and morphological type is well established, thanks to extensive spectroscopic surveys (Lin et al. 1999; Cowie, Songaila, & Hu Cowie et al. 1996; Crampton et al. 1995; Lilly et al. 1995). In particular, Wilson et al. (2001), using a large spectroscopic sample, demonstrated that objects with $(V-I)_{AB}\sim3$ are predominately massive elliptical galaxies at $z\sim0.8$. Furthermore, clustering amplitudes have recently been measured for objects selected to have extremely red colours in optical-infrared bandpasses Daddi et al. (2000). These objects have clustering amplitudes $\sim$$10\times$ higher than the full field population. It is probable that these objects are closely related to our $(V-I)_{AB}\sim3$ sample; for galaxies with (V-I)_AB>3 at I_AB<23.0, we find $\sim$0.3  $N_{\rm gal}$ arcmin^-2. In Daddi et al. (2000) using a selection of $(R-K_{\rm s})>5.0$ and the slightly brighter limit of $K_{\rm s}<19.0$, they find a surface density of 0.5  $N_{\rm gal}$ arcmin^-2. The difference between our full field clustering amplitude at 18.0<I_AB<23 and the clustering of objects selected with $(V-I)_{AB}\sim3$ is approximately the same as the difference found by Daddi et al. (2000) between their $K_{\rm s}$-selected sample and those of their extremely red objects.

Intriguingly, at the blue end of our selection, $(V-I)_{AB}\sim 0$, we also find a higher clustering amplitude than the full field sample, although the error bars are large due to field-to-field variations (at the $\sim$0.1 magnitude level) in galaxy colours and the small numbers of objects involved. We have repeated our measurement of $A_\omega (1\hbox {$^\prime $ })$ using an integrated selection (i.e., considering only objects redder or bluer than a specified colour cut) and find a similar effect. There is some evidence for this effect in the literature: working with photographic data, and considering objects in a somewhat brighter blue selected magnitude cut, $20 < B\rm _J < 23.5$, Landy, Szalay, & Koo Landy et al. (1996) also found an enhancement of at least $\sim$10 for the clustering amplitudes of the bluest objects in ($U-R\rm _F$).

Although Lyman-break galaxies are expected to be flat spectrum objects and therefore have (V-I)_AB colours of $\sim$0 their surface densities are not large enough to produce the effect seen in Fig. 18. The most likely explanation of this result is that these objects constitute a low-redshift population whose higher correlation amplitudes are a consequence of the lack of projection effects which dilute the measured $A_\omega$'s. Some evidence for this can been seen in Fig. 5 of Crampton et al. (1995); all objects with $(V-I)_{AB}\sim 0$ are at z<0.3. We also note that our red and blue samples have very low cross-correlation amplitudes, which supports the notion that the objects in our survey with $(V-I)_{AB}\sim3$ and $(V-I)_{AB}\sim 0$ are separate populations at different redshifts.

   
6.4 Biasing and the growth of galaxy clustering

Given that the redshift distribution used in our models is in agreement with our observations, then it is clear that the discrepancy evident in Fig. 20 between model predictions and observations must be a result of evolution in the intrinsic properties of the galaxy population. Our simple model does not take this into account. As a first step towards a more realistic description of the data, we may try changing the form of the r₀-z relation: McCracken et al. (2000), considered such a modification by adopting the form of $\xi(r,z)$ derived from dark matter haloes with $v_{\rm max}>120$ kms^-1 identified in a large, high-resolution N-body simulation (Kravtsov & Klypin 1999). This is shown as the solid line in Fig. 14. However, because the form of this relationship is very similar to the traditional "epsilon-model'' in the range 0<z<1, and because there are few z>3 galaxies in samples limited at I_AB<25, the differences in predicted amplitudes between this and the conventional formalism are small in the magnitude ranges we consider.

Figure 20: The evolution of $\log(A_\omega(1\hbox{$^\prime$ }))$ (assuming $\delta =-0.8,\rm\Omega _M=0.1$ and r0=4.3 h-1 Mpc) for $\epsilon =-1.2,0.0,1.0,2.0$ (long dashed, solid, dotted, and dashed lines respectively). The filled points show the CFDF full sample. This plot is similar to Fig. 14 except the literature compilation has been omitted for clarity.

The basic reason why the "$\epsilon$''-models fail to reproduce the clustering of Lyman-break galaxies and the observed form of r₀(z) at high redshift is that they implicitly ignore the existence of bias and that how galaxies trace the underlying dark matter depends on the mass of the dark matter halo (Kaiser 1984; Bardeen et al. 1986). Measurements of galaxy clustering in semi-analytic models, which provide a prescription for how galaxies trace mass, show clearly that more luminous galaxies have clustering amplitudes very different from less luminous ones (Benson et al. 2001; Baugh et al. 1999; Kauffmann et al. 1999).

Furthermore, it is now reasonably well established from observations that locally the galaxy correlation length r₀ depends on morphological type and colour (Tucker et al. 1997; Loveday et al. 1995; Davis & Geller 1976), and some evidence exists for a direct dependence between luminosity and clustering amplitude (Benoist et al. 1996). There are also indications that these trends continue to higher redshifts (Carlberg et al. 2000; Le Fèvre et al. 1996). Furthermore, in our dataset, the median field galaxy (V-I)_AB colour changes by $\sim$0.4 mag in the range 22<I_AB<24 (Fig. 11); from Fig. 18 we see that changes in colour of this magnitude cannot produce the changes in amplitude of $A_\omega (1\hbox {$^\prime $ })$ seen in the data. For this reason we suggest that the rapid decline in $A_\omega (1\hbox {$^\prime $ })$ in the range $22 < I_{AB-{\rm med}} < 24$ is a consequence of luminosity-dependent clustering segregation.

Extensive imaging and spectroscopic observations have demonstrated that, for any magnitude limited sample, as we probe to fainter magnitudes, the mean intrinsic luminosity of the field galaxy population becomes progressively fainter. This is illustrated in Fig. 21 where we show the absolute luminosity as a function of apparent magnitude for galaxies in the CFRS survey (Lilly et al. 1995) (open circles) and for galaxies in the HDF-North computed using photometric redshifts from the photometric catalogue of Yahil Fernández-Soto et al. (1999). For both these catalogues we also show the median absolute magnitudes computed in half-magnitude intervals of apparent magnitudes (open and filled circles for the CFRS and HDF respectively). We also include an estimate of the median differential luminosities ${\rm d}L/{\rm d}m$ from our luminosity evolution models (solid line). We see that between $I_{AB}\sim20$ and $I_{AB} \sim 22$, within the CFRS sample, the median galaxy luminosity declines by $\sim$0.5 magnitudes. In the range 22<I_AB<24 we measure a decline of a further magnitude, although the uncertainties in the absolute luminosities computed from photometric redshifts in the HDF is probably at least $\sim$0.5 mag. The decline in model luminosities seen in the range 18.5 < I_AB < 22.5 are a consequence of the steep faint-end slope we adopt for the spiral galaxy luminosity function. These faint galaxies at $I\sim22$ are predominately late-type galaxies, as has been demonstrated by spectral and morphological classification (Brinchmann et al. 1998; Driver et al. 1998).

Figure 21: Absolute IAB magnitudes (MI) and median absolute luminosities as a function of IAB magnitudes for galaxies selected from the CFRS (open circles, stars) and from the HDF-N (small and large filled circles). Model predictions of our luminosity evolution model are shown as the solid line (In all cases luminosities are computed assuming h=0.5 and an open $\rm\Omega _M=0.1$ cosmology).