First, we tested the reliability of our code for measuring the LS estimator of on the Zwicky catalogue. The result is consistent with that of Peebles (1974). We also used a deep catalogue (courtesy of Roukema, priv. comm. 1999) and found good agreement with its measurement.
Fig. 11 shows the result of our angular correlation code on the sample of stars brighter than . We limit the calculation of at because of the reasons invoked in Sect. 5. The function is always consistent with a random distribution. The weak positive signal might be a sign of misclassification of galaxies or simply small over-densities. The LS estimator (Eq. ) is then measured, using the entire sample of galaxies, for six cumulative magnitude intervals: , , , , , and , and four incremental magnitudes intervals: , , , and .
Random errors are computed using Eq. (23). Table 6 gives the values of and the 3- random errors for each cumulative magnitude interval, while Table 7 gives the best-fitted values of the corresponding for the parameterization of Eq. (14). Three values of the amplitude are given in Table 7. is the best-fit amplitude when both the slope and the amplitude are fitted. is the best-fitted amplitude when the slope is constrained to , and is corrected for the star dilution factor () . The amplitudes , , and correspond to the values of the correlation function at , the usual chosen reference scale. Table 7 shows the results for both the integrated intervals and incremental intervals of magnitudes. All quoted errors are 3-. For the same , the values of are consistent between the integrated intervals and the incremental intervals. Because the error bars are larger for the incremental intervals, we based our analysis on the integrated intervals only. Fig. 12 plots the LS in logarithmic scale for the I-band incremental magnitude intervals, along with the best-fit power-laws for an unconstrained slope (Table 7). Fig. 13 plots the difference of all with the constrained power-laws.
Table 6. Measures of and 3- errors (from Eq. ) in the I band for (), (), (), (), (), and ().
Table 7. Best-fit values of the amplitude and the slope of ; is obtained for a fixed slope and is the corresponding star dilution corrected amplitude (see text) in the interval . All quoted errors are . The first 6 rows of the table show the integrated intervals of magnitudes, the last 4 rows show the incremental intervals of magnitudes.
A final check on the code was done by Colombi (priv. comm. 1999) using a count-in-cells routine. The results showed that the quality of the dataset allows to measure higher orders of the distribution of galaxies (skewness & kurtosis). The measurements of the angular correlation in the interval gave consistent results with the LS estimator.
Here, we only show the correlation function in the I band. We also measured on the V-band map in the magnitude intervals , , , , and . The values of the amplitude obtained in the V band are very similar to the values obtained in I, but the average slope is in V, as compared to in I. Neuschaefer & Windhorst (1995) also measured slopes in the g and r bands and in the i band.
6.1. Variation of slope versus magnitude
Recent CDM models predict a decrease of the spatial correlation slope for scales Mpc from at to for (Kauffmann et al., 1999b). This implies a decrease of the slope of at small angular scales and faint magnitudes. Observational evidences are poorly conclusive. Brainerd et al. (1999) report a steepening of the slope on small scales while Campos et al., 1995, Neuschaefer & Windhorst, 1995, Infante & Pritchet, 1995, and Postman et al. 1998 find the opposite effect. Other authors find no significant variations to the limits of their samples (Couch et al., 1993; Roche & Eales, 1999; Hudon & Lilly, 1996; Woods & Fahlman, 1997). Fig. 13 and Table 7 show no significant flattening of slope at faint magnitudes. The slope is compatible with for all magnitude intervals. Our result is nevertheless consistent with the results of Postman et al. (1998) who find signs of a decrease only in their faintest bins, at , near and beyond our I limit.
6.2. Variation of with magnitude
The choice of a galaxy luminosity function to model the decrease of with the limiting apparent magnitude is crucial because the behavior of is sensitive to both the parameterized characteristic absolute magnitude and the slope (Eq. ). We choose to use the luminosity function observed in the CFRS (Lilly et al. 1996), derived from 591 galaxies in the range , keeping in mind that such a small sample of galaxies can only provide an indication of a general trend. Because both the CFRS sample and our sample have been selected in the I band, we thus limit the possible biases due to the photometric sample selection.
Lilly et al. divide the CFRS sample into a red population (redder than an Sbc having rest-frame ; at , , and ) and a blue population (bluer than an Sbc). The red galaxies show no density or luminosity evolution in the range , whereas the blue galaxies show a luminosity evolution of about 1 magnitude in the same redshift range. Recent measurements made from the CNOC2 survey (Lin et al., 1999), on 2000 galaxies in the range , confirms the general observations of the CFRS, although the proposed interpretation is notably different. The CNOC2 analysis separates the luminosity evolution from the density evolution. Early and intermediate (red) galaxies show a small luminosity evolution in the range (), whereas late (blue) galaxies show a clear density evolution with almost no luminosity evolution in the same redshift range, in apparent contradiction with the CFRS results.
We choose to adapt the CFRS LF to our sample, and we proceed as follows. Galaxies are separated into two broad spectro-photometric groups, the E/S0/Sab (called red) and the Sbc/Scd/Irr (called blue), using the median colour (see Sect. 4.3). The red group has a non-evolving luminosity function with parameters Mpc-3, , and (Eq. ), and the blue group has a mild-evolving luminosity function with parameters Mpc-3, , and . The factor equals 0 for and for and mimics the observed smooth brightening of with redshift.
We integrate Eq. (18) in the different apparent magnitude intervals listed in Table 6, and obtain for three cosmologies. The K corrections are computed from templates of E (9 Gy) for the red group and Sd (13 Gyr) for the blue group from the PEGASE atlas (Fioc & Rocca-Volmerange, 1997) and are shown in Fig. 14. The choice of a given atlas is not benign, as Galaz & de Lapparent (1998) shows that K corrections can vary by 50% at when comparing the PEGASE atlas with the GISSEL atlas (Bruzual & Charlot, 1993), leading to significant differences in .
Fig. 15 shows for red and blue objects having , for three cosmologies: Einstein-de Sitter as a solid line, open universe as a dashed line, and flat universe as a dotted line. One can note that red and blue galaxies show very different redshift distributions as expected from the different LFs. This should be kept in mind when we compare the different evolution of with magnitude for the red and blue samples. The resulting used to model our sample is the sum of the for the red and blue object distributions respectively. As our 2 colour samples suffer from differential incompleteness (see Sect. 3.3), we normalized the relative number of blue and red objects to the observed ratio in the CFRS survey at the corresponding limiting magnitude.
Fig. 16, Fig. 17, and Fig. 18 show the decrease of the amplitude of the correlation function corrected for star dilution for the median I magnitude of the integrated and incremental intervals (Table 7); a fixed slope of is used to measure the reference scale, which is taken a 1 degree. The data-points for the incremental intervals show larger error bars, but are consistent with the amplitudes for the integrated intervals. The measurements of Postman et al. (1998) and Lidman & Peterson (1996) are shown for comparison. We also plot the expected curves for the three universes mentioned above, for each of the three values of the clustering parameter in Fig. 16, in Fig. 17, and in Fig. 18). For each value of , the theoretical curves are calculated with and the best-fit spatial correlation length at , using Eqs. (15) and (22). These values along with the corresponding of the fit are listed in Table 8 (only the integrated intervals of magnitude are used for the fitting, 6 data-points). Note that in Fig. 16, Fig. 17, and Fig. 18, different values of shift the theoretical curves by constant values in the Y direction.
Table 8. Best-fit values of the spatial correlation length for fixed and three cosmologies.
Several conclusions can be drawn from Fig. 16, Fig. 17, Fig. 18, and Table 8. We can always find a set , and which fits our data. universes with and slightly favour null , in good agreement with the results of Baugh et al. (1999) for semi-analytical models of biased galaxy formation. In contrast, Table 8 shows that Einstein-de Sitter universes favour positive (recall that means no evolution in comoving coordinates, and no evolution in physical coordinates), as obtained by Hudon & Lilly (1996) () for the hierarchical clustering CDM model of Davis et al. (1985) in Einstein-de Sitter Universes. Third, positive values of evolution parameter give better fits to our observations (Fig. 18) than negative . Moreover, comparison of the listed in Table 8 with the local values of Mpc at also suggest null to mild clustering evolution.
Finally, Table 9 gives the peak redshift derived from the redshift distributions (using the CFRS model luminosity function) for all magnitude intervals and for the three cosmologies. Using Eq. (22), the values of are computed for all , and can be compared to other results. At , we measure a value of in the range Mpc, depending on the cosmological model and the evolution index . The typical 1- uncertainty on the values of listed in Table 9 is Mpc. The additional error originating from the uncertainty in the cosmology and in is estimated from Table 8 and Table 9 to be Mpc. By adding these errors in quadrature, we find an estimated total error in the correlation length of Mpc. This more realistic error can also be applied to the values of given in Table 8.
Table 9. Redshift of the peak of the redshift distribution and the corresponding best-fit correlation lengths for different magnitude intervals and cosmologies. Here
If we assume a flat cosmology, Table 9 gives Mpc at the peak redshift of the survey. The corresponding value at is Mpc. Within the error bar, this result is in agreement with most other angular correlation measurements at (Postman et al., 1998; Hudon & Lilly, 1996; Roche & Eales, 1999; Woods & Fahlman, 1997).
If we compare with the direct spatial measurements, our result is closer to the CNOC2 results than those from the CFRS: Carlberg et al. find that the 2300 bright galaxies in the CNOC2 survey (Yee et al., 1996) show little clustering evolution in the range with Mpc at , depending on the cosmological parameters (Carlberg et al., 2000); whereas Le Fèvre et al. measure Mpc at for 591 I-selected galaxies, implying a strong evolution from the local values ( Mpc at ) (Lilly et al., 1995). As mentioned above, the small correlation length found in the CFRS survey may be due to the cosmic variance which affects small area surveys. On the other hand, neither the CNOC2 survey nor the CFRS survey detect an evolution in the slope , in good agreement with our UH8K data.
6.3. Variation of with galaxy colour
We also calculate the variations of with depth for the blue and red sub-samples of our UH8K data, which are defined in Sect. 4.3. A map of the 8986 red galaxies () with is shown in Fig. 19 (top panel); the 7259 blue galaxies () to the same limiting magnitude are plotted in the bottom panel of Fig. 19. One can see that red objects are more clustered than blue objects, as partly reflected by the morphology density relationship (Dressler, 1980). Note that the surface density of the blue galaxies increases by a factor of with increasing right ascension. This is probably caused by a systematic drift in the photometric zero-point along the survey R.A. direction, which was not completely removed by the matching of the magnitudes in the CCD overlaps (see Sect. 3.2). We then perform an angular correlation analysis on each colour-selected sample. For the blue sample, we introduce in the random simulations, prior to masking, the mean R.A. gradient measured from the data.
Fig. 20 shows the decrease of for the red and blue galaxies to a limiting magnitude (see Sects. 3.3 & 4.3). In principle, the separation should be done on rest-frame colours and not on observed colours, but this requires prior knowledge of the redshift of the objects. The effect of using observed colours is to decrease the resulting angular correlation because galaxies of different intrinsic colours at different distances are mixed together. We use the same angular binning as in the previous analyses for consistency (Sects. 6.1 & 6.2). Table 10 gives the median magnitude of the red and blue samples. The last interval given in Table 10 includes all 11,483 red galaxies () and 20,743 blue galaxies () fainter than (to and ).
Table 10. Median magnitude of the red and blue samples for the different cumulative I magnitude intervals.
Note that the blue sample reaches fainter I magnitudes than the red sample. As seen in the Sect. 3.3, this is a selection effect due to the fact that only galaxies detected in the 2 bands are shown in the sample, and the blue sample is complete to . Thus at , the galaxies redder than (in fact most of the red sample) show sparse sampling. Each interval in Table 10 is respectively complete for galaxies having . We emphasize that the last two intervals of the red sample ( and ) are probably too incomplete to be considered as fair samples. The incompleteness induces two competing effect. On the one hand, it increases , because the number of objects is artificially small and the resulting correlation is higher; on the other hand, it makes the median magnitude brighter (cf Table 10), thus inducing a steepening of the slope of the decrease. In Fig. 20, the obviously erroneous offset of last point for the blue sample, corresponding to the full magnitude range and a median I magnitude of , also illustrates the error on the measurements of one can expect in extreme cases where magnitudes are poorly defined and objects are under-sampled.
Keeping in mind the warning of the previous paragraphs, and the fact that the error bars prevent us from drawing strong conclusions, it is striking to see the different amplitude of for the two colour samples shown in Fig. 20. The larger for red galaxies than for blue galaxies for all is in agreement with the well-known higher clustering amplitude of early-type galaxies (see for example recent spatial measurements in the Stromlo-APM survey by Loveday et al., 1995), which is partly reflected by the morphology density relationship (Dressler, 1980). Using the first three data-points of the red sample and blue samples in Fig. 20, we calculate that the two distributions are different at a 4- level. The corresponding clustering amplitudes are Mpc for the red sample and Mpc for the blue sample; both are based on a universe with . For the purpose of comparing these values of , the mentioned errors assume that the correlation functions for the red and blue galaxies have the same slope . This may not be true (see Loveday et al. 1995), but our UH8K sample does not allow to address this issue. We also ignore the systematic effect of the cosmology which would affect the values of for the two populations in the same direction. The resulting difference in the measured for the red and blue populations are at the 3- level.
Note that if the clustering of the red and blue galaxies were identical, the larger average distance for the red galaxies compared to the blue galaxies (see Fig. 15) would yield lower for the red galaxies than for the blue galaxies. The opposite effect is observed. Moreover, if the larger number of red galaxies (in the first 4 magnitude intervals in Fig. 20) in the red sample (8986) compared to the blue sample (7259, see Table 10) is caused by large random errors at the limit of the catalogues, as seen in Sect. 5.3.2, these would tend to dilute the for the red sample. The detected increased clustering strength for the red sample over the blue sample is therefore a lower limit on the amplitude difference with colour. The apparent flattening of with for red galaxies in Fig. 20 may be due to the incompleteness of red objects at faint magnitudes as discussed above.
The difference in clustering amplitudes which we measure for our red and blue samples agrees with observations by Neuschaefer et al. (1995), Lidman & Peterson (1996) and Roche et al. (1996). Neuschaefer et al. find that disk-dominated galaxies (blue ) have marginally lower than bulge-dominated galaxies (red ) using HST multi-colour fields. Similarly, Roche et al. observe a 3 difference in for a sample divided into objects bluer or redder than . Lidman & Peterson see a weak difference between two samples separated by . Other authors don't see any difference between blue and red-selected samples, such as Woods and Fahlman (1997) for a separation of , Brainerd et al. (1995) for , Le Fèvre et al. (1996) in the spatial correlation length of the CFRS for rest-frame , and Infante & Pritchet (1995) for . In all these cases excepting Infante & Pritchet, who used photographic plates, both the number of galaxies and the angular scale of the surveys are small. It might be possible that in these surveys cosmic variance hides a weak signal.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001