7.1. Limber's formula
Limber's formula (written here as Eqs.  and ) relating and is strongly dependent on the shape of the redshift distribution which depends on the characteristic absolute magnitude and slope of the luminosity functions of the different types of galaxies (Eq. ). Locally, these parameters cover a wide range of values with regard to the environment and the morphological types of the galaxies (Binggeli et al., 1988; Lin et al., 1999; Marzke et al., 1998; Bromley et al., 1998; Galaz & de Lapparent, 2000). No reasons lead us to think that galaxies would show less diversity at higher redshifts. Recent observations (in UV, IR, X, and ) of space-borne observatories do provide evidence in this direction as well as the discordant measurements of the LF by the CFRS and CNOC2. Hence, a prior knowledge of the detailed luminosity functions is necessary to go beyond a phenomenological description of the clustering of galaxies.
Our computed is credible only if the correct luminosity functions have been used in Limber's formula: this would guarantee that the modeled is close to the true redshift number distributions for each galaxy type. Segregating galaxies into red and blue samples based on observed colour, as we do here, is also a crude first step, and should rather be performed using intrinsic colour or, even better, spectral type (cf Sect. 6.3). These in turn would require knowledge of the galaxy redshifts. Approximate redshifts can also be obtained along with spectral type for multi-band photometric surveys using photometric redshift techniques (Koo, 1999). Because none of the required functions and distributions are available for our UH8K sample, we emphasize that the reported results can only be taken as phenomenological, and all the comments on the deduced clustering must be taken with caution.
7.2. The cosmological parameters
Most authors take for granted that different cosmological parameters only lead to minor differences in the evolution of clustering, compared to the effects due to the uncertain luminosity function. Fig. 16 to Fig. 20 do show that for a given luminosity function different cosmological parameters induce different values of and . From our models, differs by more than between the flat and Open universes. These differences cannot yet be distinguished with the present data (up to ). The dispersion between the different observations (Fig. 21) precludes any derivation of the cosmological parameters.
Given a CFRS luminosity function, the flat universe gives the better fits to galaxy number counts, and clustering evolution of our sample (). This is in agreement with a current (though controversial) interpretation of recent type Ia supernovae results (Schmidt et al., 1998; Perlmutter et al., 1999). Notwithstanding the numerous modern methods to measure the cosmological parameters, the present analysis shows that future surveys containing galaxies with known luminosity functions per galaxy type and redshift interval to will be required to provide good constraints on cosmological parameters using this technique.
7.3. The evolution of clustering
Fig. 21 shows the decrease of the amplitude of the angular correlation of our sample compared to most of the other recent measurements made in the I band (Brainerd & Smail, 1998; Campos et al., 1995; Woods & Fahlman, 1997; Postman et al., 1998; Lidman & Peterson, 1996; Neuschaefer & Windhorst, 1995; McCracken et al., 2000b). We applied a magnitude translation between Neuschaefer & Windhorst i magnitude and our I magnitude of .
In the magnitude range I=20-22, our results are in good agreement with most of these results except those of Campos et al., which have the highest amplitude, and on the low side, those of Lidman & Peterson and of McCracken et al., 3 times lower in amplitude. Postman et al., Woods et al., and Neuschaefer & Windhorst obtain intermediate values at .
Postman et al. observe a flattening of for scales and . The measurements of Brainerd & Smail extend the flattening of observed by Postman et al. to . We also find a possible flattening in the decrease of beyond . The most probable explanation for such high dispersion of about a factor ten between the different measurements of is the dependence of on the square of the number density. Cosmic variance may account for most of the discrepancies. The rest may be attributable to systematic errors of the estimators due to different spatial or magnitude samplings.
As pointed out by Neuschaefer & Windhorst (1995), a flattening of the slope (see Eq. ) with the redshift or with apparent magnitude would lower the theoretical curve for derived with Limber's formula (Eq. ). Hence, smaller would still be compatible with a small , or a high . In other words, clustering would grow faster at smaller scales than at larger scales. The flattening of the spatial correlation function is predicted by N-body simulations (Davis 1985; Kauffmann 1999a; 1999b). Neuschaefer & Windhorst parameterize the flattening of the slope with redshift z as: , where . Postman et al. (1998) derive with a slightly different parameterization: .
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001