5.2 Measurements of $\omega (\theta )$ in the CFDF fields

   
5.1 Measuring $\omega (\theta )$

There is an extensive literature on the measurement of the projected galaxy correlation function in deep imaging data (Efstathiou et al. 1991; Roche et al. 1993; Brainerd et al. 1994; Hudon & Lilly 1996; Woods & Fahlman 1997; McCracken et al. 2000). Here we will only briefly outline the relevant equations. We have computed the two-point projected galaxy correlation function using the Landy & Szalay (1993) (LS) estimator,

$$\omega ( \theta) ={{DD} - 2{DR} + {RR}\over {RR}},$$ (3)

with the DD, DR and RR terms referring to the number of data-data, data-random and random-random galaxy pairs between $\theta$and $\theta+\delta\theta$. In this work we use logarithmically spaced bins with $\log(\delta\theta)=0.2$.

The fitted amplitudes quoted in this paper assume a power law slope for the galaxy correlation function, $\omega(\theta)=A_\omega\theta^{-\delta}$; however this amplitude must be adjusted for the "integral constraint'' correction, arising from the need to estimate the mean density from the sample itself. This can be estimated as (Roche et al. 1993),

$$C = {1 \over {\Omega^2}} \int \int \omega(\theta) {\rm d}\Omega_1 {\rm d}\Omega_2,$$ (4)

where $\Omega$ is the area subtended by each of our survey fields. For the CFDF fields, we find $C\sim 4A_\omega$ by numerical integration of Eq. (5) over our field geometry and assuming that galaxies closer than 1'' cannot be distinguished. In deriving the correlation amplitudes we then fit

$$\omega_{\rm obs}(\theta) = A_\omega \theta^{-\delta} - C,$$ (5)

where $\omega_{\rm obs}(\theta)$ is our observed measurement of $\omega (\theta )$. (We note however that the integral constraint correction only becomes important at larger scales ( $\theta > 2\hbox{$^\prime$ }$) and that the large numbers of galaxies in our survey combined with the large field of view of each patch means we can fit for $A_\omega (1\hbox {$^\prime $ })$ while neglecting the integral constraint, providing the range of the fit is restricted to scales where the integral constraint is not important). In our fitting procedure we use the method of Marquardt (1963) which takes into account that each measurement of $\omega (\theta )$ at each angular separation is not independent of the others.

To minimise computational requirements (necessary given the large numbers of galaxies in our samples, typically 30000 galaxies per field) we use a sorted linked list method to compute the numbers of galaxy pairs in each angular bin (and furthermore we have verified that this method gives the same results as a direct-pair counting approach; it is, however $\sim$ $2{-}3 \times$ faster). We use polygonal masks to blank out regions surrounding bright stars, large galaxies, satellite trails and cosmetic defects. We also discard data from CCD 8 because of its poor charge transfer properties. Our masking strategy is quite conservative, and (excluding CCD 8), around $15{-}20\%$ of the total area is removed. Because the 11 hr and 22 hr fields are composed of two separate stacks which are rotated relative to each other, are able to cover the entire $28'\times28'$ area for these fields. We have verified that the masks do not bias the determination of $\omega (\theta )$ by applying them to randomly generated catalogues with the same number density as the real catalogues and verifying that $\omega (\theta )$ is zero at all angular scales. We have also tested the effect of masks on clustered catalogues generated using the method of Soneira & Peebles (1978). These tests (which were carried out using the LS estimator) show that the application of masks does not change our fitted correlation amplitudes. Note that we do not apply a stellar dilution correction to our measured correlation amplitudes as some authors do, as the form of the stellar counts for high galactic latitude fields is not precisely known. However, this correction is unlikely to raise the amplitude of the correlation function by more than $10{-}15\%$, given that stars outnumber galaxies by at least an order of magnitude beyond  $I_{AB}\sim21.5$.

   
5.2 $\omega (\theta )$ for I $_{\mathsfsl{AB}}$ magnitude-limited samples

For each of our four fields, we divide our catalogue into magnitude limited samples. Each sample reaches one half-magnitude deeper than the previous, while the bright limit is kept at I_AB=18.5. (As the 14 hr field is shallower than the other fields, we do not measure  $\omega (\theta )$ on this field fainter than I_AB=24.0.) These samples are extracted from the $\chi ^2$ catalogues prepared as outlined in Sect. 3.1; we have also performed the same computation on single-band catalogues (i.e., those computed without using an associated $\chi ^2$ image for detection) and find identical results. Fig. 12 shows the logarithm of the amplitude of $\omega (\theta )$ averaged over the four fields of the CFDF as a function of the logarithm of the angular separation in degrees, for a number of magnitude-limited samples. In the range $-2.5< \log (\theta) < -1.3$ ( $0.2' <\theta < 3'$) our measurements follow the expected power-law behaviour very well and at least until $I_{AB}\sim24$ there is no evidence of significant excess power on scales larger than the individual UH8K CCDs ($\sim$10'), in agreement with the evidence presented in Sect. 2.5 that the systematic photometric errors in our fields are negligible.

Figure 12: The logarithm of the average amplitude of the angular correlation function, $\omega (\theta )$, as a function of the logarithm of the angular separation in degrees, for all fields in the CFDF. The error bar on each bin is computed from the field-to-field variance. Each of the five different symbols shows a different magnitude-limited sample; the solid lines shows the fitted correlation amplitude. This weighted fit is carried out neglecting the three innermost bins and assuming a power-law slope of -0.8 for  $\omega (\theta )$ and a value of 4.2 for the integral constraint term C. Data from the 14 hr field is not included in the faintest slice.

Figure 13: Simulated measurements of $\omega (\theta )$, b,c), compared with CFDF measurements for slice 18.5<IAB<24.0, a). In each case the simulated measurements have been normalised by the fitted amplitude for the 18.5<IAB<24.0 slice, taking into account the integral constraint correction and assuming a $\delta =-0.8$ power law. The simulations b,c) contain objects in the magnitude range 23.5<IAB<24.5 and 24.5<IAB<25.5 respectively. Each plot has been offset by an arbitrary amount x where x=0,1.5,3.0.

At $\theta < 0.2'$ we find at fainter magnitudes that $\omega (\theta )$deviates from the expected power-law behaviour. We have attempted to investigate the origin of this effect (which is seen in all our fields) by carrying out an extensive set of simulations. In these simulations we generate a catalogue with $\sim$$40\,000$ objects which have the same correlation amplitude as the $I_{AB}\sim24$ sample. Each of these objects is assigned a random magnitude in a specified interval and then added to the data frame. Object detection and photometry is carried out in a box extracted at this location. A new catalogue is constructed containing recovered total magnitudes for each object, and a magnitude cut is then applied to this sample. Objects which are lost in this process are typically those falling on or near bright stars or galaxies, or those whose recovered total magnitude falls outside our magnitude cut. Masks are applied to this catalogue and $\omega (\theta )$ computed using the same procedures used for the real data. This procedure is then repeated for progressively fainter magnitude slices. (In constructing the simulated catalogue, two simplifying assumptions were made, firstly that the input magnitude distribution of objects is flat, and secondly we use objects with Gaussian point-spread functions.)

In Fig. 13 we show the results of two of these simulations, labelled (b) and (c). The results displayed for (a) show the measured correlations for the 18.5<I_AB<24.0 magnitude slice. In order to display deviations from the power-law behaviour, each slice has been normalised by the fitted amplitude for the 18.5<I_AB<24.0 slice, taking into account the integral constraint correction and assuming a $\delta =-0.8$ power law. These simulations cannot reproduce the depression in the correlation function found at scales $\log(\theta)\sim -2.5$. We have also investigated if the depression is caused by an excess of objects around bright stars, and have found no evidence of such an excess. An important aspect of these simulations, however, is that they confirm that excess power on large scales only becomes important for the CFDF data for the faintest magnitude ranges, 24.5<I_AB<25.5.

Figure 14: The logarithm of the fitted amplitude of the angular correlation  $\omega (\theta )$ at 1' as a function of sample median IAB magnitude (filled circles). Fits were computed assuming a power law of slope $\delta =-0.8$ for  $\omega (\theta )$ and an integral constraint term calculated as described in the text. Error bars on the CFDF points are computed from the analytic expression of Bernstein Bernstein Bernstein 1994. The model curves are taken from McCracken et al. (2000) and show three different clustering evolution scenarios which are detailed in the text.

In Fig. 14 we plot the fitted amplitude of $\omega (\theta )$ at one arcminute, $A_\omega (1\hbox {$^\prime $ })$, averaged over all our fields, (filled circles) as a function of the sample median ABmagnitude ( $I_{AB-{\rm med}}$). We have estimated the error bars on $A_\omega$, $\delta A_\omega$, using an analytic approximation introduced by Bernstein (1994) and further developed by Colombi (Szapudi & Colombi Szapudi & Colombi 1996); see also Arnouts et al., in preparation for a detailed explanation of the application of this approximation. The Bernstein estimator relies on a knowledge of the higher-order moments of the galaxy correlation function, S₃ and S₄; we have estimated these quantities directly from the CFDF dataset and they will be presented in a future work (Colombi et al., in preparation). At bright magnitudes ( $I_{AB-{\rm med}}\sim 18$), $\delta A_\omega$ is dominated by the Poisson error component; however, faintwards of $I_{AB-{\rm med}}\sim 22$, the main component of  $\delta A_\omega$ in our field consists of cosmic variance (or "finite volume'') effects. Our analytic estimates of this effect indicate that the errors estimated empirically from the field-to-field variance of our four fields may underestimate the total error at these magnitudes by around $\sim$$20\%$.

Our fitted amplitudes were computed assuming a -0.8 slope for $\omega (\theta )$ (to allow a comparison with the literature), and over the range $0.2' <\theta < 3'$. We have also attempted to fit for $A_\omega$ only at larger separations ($\sim$ $3\hbox{$^\prime$ }$), but our survey fields are not large enough to detect any scale-dependence in the $A_\omega-I_{AB-{\rm med}}$ relation. Additionally, Fig. 14 shows a compilation of recent measurements of $A_\omega$ from the literature. These literature measurements all assume a fixed slope of $\delta =-0.8$, with the exception of the Postman et al. survey, in which the slope was allowed to vary with the fit. To allow a fair comparison with the other authors, and with our work, the Postman et al. points are not plotted faintwards of $I_{AB-{\rm med}}\sim 21$, where their fitted slopes begin to differ significantly from -0.8.

At bright magnitudes ( $19 < I_{AB-{\rm med}} < 22$) we find that our observations are compatible with almost all the data from the literature compilation. At fainter magnitude ranges, ( $22 < I_{AB-{\rm med}} < 24$), our observations clearly favour a low amplitude for $A_\omega$. At $I_{AB}\sim24$ they are least a factor of ten below the value of $A_\omega$ found by Brainerd & Smail (1998). This work covers a small area ($\sim$50 arcmin²) and consists of individual unconnected pointings. As has been suggested before (McCracken et al. 2000) the discrepancy may be due to field-to-field variance in the galaxy clustering signal. To provide a more direct answer this question, we extract 200 regions of 50 arcmin² from the 22 hr and 11 hr fields (these two have complete coverage as a consequence of bonette rotations). On each of these sub-regions we measure $A_\omega$ as for the full sample. In Fig. 15 we show the histogram of the fitted values for the 11 hr and 22 hr fields; from this we estimate that a $\pm 3\sigma$ error bar indicates a dispersion of $\times 10$ in $A_\omega$. It is clear that the error bars on Brainerd & Smail measurement underestimate the true error by a large amount.

Figure 15: The logarithm of the fitted correlation amplitude at  $1\hbox {$^\prime $ }$, $\log(A_\omega)$ measured on 200 $7\hbox {$^\prime $ }\times 7\hbox {$^\prime $ }$ sub-areas extracted from the 11 hr and 22 hr fields. Galaxies extracted have magnitudes in the range 18.5 < IAB < 25.5. The dotted lines represent  $\pm 3\sigma$ confidence limits about the mean value, shown by the solid line. The arrow shows approximately the $A_\omega (1\hbox {$^\prime $ })$ measured by Brainerd & Smail (1998).

We have also determined $A_\omega (1\hbox {$^\prime $ })$ in one-magnitude slices, for instance, 19.5<I_AB<20.5, 20.5<I_AB<21.5 to the limit of our survey, as an additional check. These measurements are considerably more noisy than the integrated measurements presented above because of the smaller numbers of galaxies in each slice. However, we find that the derived $I_{AB-{\rm med}}-\log A_\omega(1\hbox{$^\prime$ })$ relation is very similar to what is presented in Fig. 14.

   
5.3 Dependence of slope on magnitude

The large numbers of galaxies in our survey allows us to make a direct measurement of $\delta$, the slope of  $\omega (\theta )$ as a function of sample limiting magnitude. Several attempts at this measurement have been carried out but with generally inconclusive results. Neuschaefer & Windhorst (1995) find $\delta=-0.5$ at g<25 based on two independent fields each the size of one of our CFDF fields. Shallower wide-angle surveys like those of Roche & Eales (1999) and Cabanac et al. (2000) find no evidence for deviation from a slope of $\sim$0.8. One difficulty in this measurement is that excess power on large scales, such as can be produced by zero-point variations, can produce an artificially shallow slope. However, we have already demonstrated that our magnitude zero-point errors are small across our mosaics (Fig. 4) and that differential incompleteness only becomes significant for measurements of  $\omega (\theta )$ in the CFDF fields at very faint magnitudes (Fig. 13). Nevertheless, we adopt a cautious approach and use two different methods to measure $\delta$.

Firstly we compute $\chi ^2$ contours from the average correlation per angular bin over all the fields (Fig. 12), shown in Fig. 16. In our second method we compute the slope independently for each field and measure the field-to-field standard deviation of this fit, which is shown in Fig. 17, together with a comparison with the fitted slopes from Postman et al. (1998), computed for a fitting range of $0.5\hbox{$^\prime$ }<\theta<5\hbox{$^\prime$ }$. An additional complication is that in both cases our slope-fitting procedure requires an estimation of the integral constraint (Eq. (5)), which in turn depends on the slope. To limit these difficulties, we perform the fit only in the range $-0.2\hbox{$^\prime$ }< \theta< 1.9\hbox{$^\prime$ }$ where the effects of the integral constraint are expected to be negligible.

Both methods indicate that at bright magnitudes, 18.5<I_AB<22.0, $\delta\sim -0.8$; at fainter magnitudes we detect a slight flattening of the slope, with $\delta \sim -0.6$. The error bars in Fig. 17 show the error in $\delta$ for a given value of $A_\omega$; from Fig. 16 we see, however, that a slope of $\delta =-0.8$ is still within $3\sigma$ of our best fit for all magnitude ranges.

Figure 16: $\chi ^2$ contours showing best-fitting amplitudes and slopes (plus symbols) for the four faintest CFDF samples, from right to left, 18.5<IAB<22,23,24,25. The three contours show the $1\sigma$ (thick contour), $2\sigma$ and $3\sigma$ confidence levels.

Figure 17: The fitted slope of $\omega (\theta )$ as a function of sample median magnitude (open circles) for galaxies with 18.5<IAB<22,23,24,25 averaged over the CFDF fields. Points are also shown from Postman et al. (1998) (filled circles). Error bars on the CFDF points are computed from the variance over our four survey fields.

   
5.4 Galaxy clustering as a function of colour

For all galaxies in our sample we can measure (V-I)_ABcolours and we can use this to select subsamples by colour. In Fig. 18 we show $\log A_\omega (1\hbox{$^\prime$ })$ as a function of (V-I)_AB colour for galaxies with 18.5<I_AB<24.0(open pentagons) and 18.5<I_AB<23.0 (asterisks). For each of the four fields, we divide our sample into twelve equally spaced bins in colour each of width 0.25 in (V-I)_AB. The error bars in $\log A_\omega (1\hbox{$^\prime$ })$ were computed from the field-to-field variance. Also shown as the dotted and dashed lines is the amplitude of $\log A_\omega (1\hbox{$^\prime$ })$ for the full-field sample for these two slices.

We first note that both these cuts are relatively bright; at $I_{AB}\sim24$ our catalogues are expected to be $\sim$$100\%$ complete (Figs. 6, 9). Additionally, as shown in Fig. 11, the effect of colour incompleteness should be minimal, although at $I_{AB}\sim24$ we may begin to lose some extremely red objects from our sample.

Our measurements clearly show that redder objects are more strongly clustered, as has been widely reported for local populations (Loveday et al. 1995). We find that objects with $(V-I)_{AB}\sim3$ have a clustering amplitude at least a factor of ten higher than the full field population, shown as the dotted and dashed lines in Fig. 18. Interestingly, we also find that the bluest objects in our survey, those with $(V-I)_{AB}\sim 0$, have a clustering amplitudes marginally higher than the full sample. Furthermore, we find that none of our colour selected samples have clustering amplitudes below the full-field value. We discuss the implications of these results in the following section.

Figure 18: The logarithm of the fitted amplitude of the angular correlation $\omega (\theta )$ at 1' as a function of sample median IAB magnitude (as before, fits assume $\delta =-0.8$). $\log(A_\omega)(1\hbox{$^\prime$ })$ was computed for 12 equally spaced bins of 0.25 in (V-I)AB. Error bars for each colour slice are computed from the field-to-field variance. Two magnitude ranges are shown, 18.5<IAB<24.0 (open pentagons) and 18.5<IAB<23.0 (asterisks), together with the fitted amplitude for the full field samples, dotted and dashed lines respectively.