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Astron. Astrophys. 364, 349-368 (2000)

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5. Analysis

5.1. Estimation of [FORMULA]

The 2-point angular correlation function [FORMULA] is calculated by generating samples of random points covering the same area and having the same number as the galaxy sample. We use the estimator [FORMULA] defined by Landy & Szalay (1993) (hereafter LS), which has the advantage of reduced edge effects and smallest possible variance:


where DD is the number of galaxy-galaxy pairs, DR the number of galaxy-random pairs, and RR is the number of random-random pairs, all of a given angular separation [FORMULA]. Following Roche et al. (1996) and Roche & Eales (1999), we set a logarithmic binning for the separation defined as [FORMULA]. The numerical approach of LS is used to calculate DD, DR and RR. If one defines the variables d and x as



then Eq. (8) can be re-written as


[FORMULA] is the probability of finding two randomly placed galaxies separated by a distance [FORMULA], n is the number of real galaxies in the sample, and [FORMULA] is the average over N realizations of a random sample of k points, with [FORMULA] (k varies between 10 to 100 depending on the magnitude interval in which the angular correlation function is measured). LS also define [FORMULA], the probability of finding two neighbors both at a distance [FORMULA] of one given object.


where [FORMULA] is the average number of unique triplets. [FORMULA] is necessary to evaluate the random errors (cf Sect. 5.3).

5.2. Modeling of [FORMULA], [FORMULA] and [FORMULA]

The canonical parameterizations of the two-point spatial correlation function [FORMULA] (Phillipps et al., 1978) and of the angular correlation function [FORMULA] (Peebles, 1980) are


where r is the comoving distance, [FORMULA] the correlation length at [FORMULA], [FORMULA] the angular separation in radian, [FORMULA] is the amplitude of angular correlation function, [FORMULA] and [FORMULA] are the slopes ([FORMULA]) and [FORMULA] is a parameter characterizing the evolution of clustering with the redshift z. If [FORMULA] the clustering evolves in proper coordinates, if [FORMULA] the clustering is constant in proper coordinates hence increases in an expanding universe, if [FORMULA] the clustering is constant in comoving coordinates. The comoving correlation length at a redshift z is related to [FORMULA] by


Given Eq. (14) and the galaxy redshift distribution [FORMULA], one can relate [FORMULA] and [FORMULA]:


Here [FORMULA] is the angular diameter distance, [FORMULA] is the comoving volume element, both given for three cosmologies in the appendix of Cole et al. (1992). [FORMULA] is the luminosity function, whose definition might be dependent on different spectral type i evolving with z, and [FORMULA] is the gamma function. The value of C is given for a typical value of [FORMULA] (Peebles, 1980). Eqs. (16) to (21) allow to make a direct derivation of [FORMULA] from [FORMULA], where [FORMULA] if the peak of the redshift distribution of the galaxies in the sample defined by the interval of apparent magnitude [FORMULA]:


([FORMULA] is rewritten as [FORMULA]).

5.3. Estimation of errors

In the past few years, considerable theoretical efforts have been devoted to the calculation of errors in the estimation of [FORMULA]. The errors can be divided into two categories; (1) the random errors; and (2) the systematic errors due to the various observational biases in the data. The systematic errors are caused by false detections, star/galaxy misidentifications, photometric variations and astrometric errors. The random errors are induced by the finite area of the survey and depend on the geometry and the size of the sample.

5.3.1. Random errors

Until recently, a Gaussian approximation for the distribution of the galaxies was assumed to calculate the errors on [FORMULA]. However, the distribution of the galaxies is known to depart significantly from a Gaussian distribution on small scales. A finer approach would be to include the possible correlations of the data into the error analysis. This has been done by Bernstein (1994) on the LS estimator [FORMULA]. Bernstein derives a formal solution to the random errors for the case of a hierarchical clustering universe in the limits [FORMULA] (number of galaxies in the sample), [FORMULA], and [FORMULA] angular size of the sample:


where the parameters [FORMULA] and [FORMULA] are measured by Gaztañaga (1994). They are related to the hierarchical amplitudes [FORMULA] and [FORMULA] ([FORMULA], where [FORMULA] are the n-point angular correlation functions), by [FORMULA] and [FORMULA] (Gazta"naga, 1994). [FORMULA] and [FORMULA] have been derived by Gaztañaga (1994) and Roche & Eales (1999) from the APM catalogue (Maddox et al., 1990a): [FORMULA] and [FORMULA] at [FORMULA], the scale at which the angular-correlation function is well measured in the APM catalogue (Maddox et al., 1990b); therefore [FORMULA] and [FORMULA]. [FORMULA] (r for ring) is the 3-point angular correlation for triplets of galaxies defined by 2 galaxies at a distance in the interval [[FORMULA],[FORMULA]] from the 3rd galaxy. In principle, [FORMULA] is marginally greater than [FORMULA] as defined in Eq. (8) but [FORMULA] is a fair approximation. [FORMULA] is the average value of the angular correlation function at the largest angular separation of the sample. This quantity is not directly accessible because the estimator W is biased by finite volume errors at separations similar to the size of the survey. To get an estimation of [FORMULA] we divide our sample into 8 sub-samples (one separation at mid-declination and three separations in right ascension) and measure the LS estimator W. Then, assuming a power-law whose slope is set at small angles we extrapolate the value [FORMULA] for each sub-sample. Finally, [FORMULA] is obtained as the average over the 8 sub-samples. Subdiving the sample into 8 sub-samples also allows to measure the error on W independently at all scales (the largest angular separation being a quarter of the largest full survey separation), by simply taking the variance over the 8 sub-samples. The resulting variance largely underestimates the variance defined in Eq. (23).

5.3.2. Systematic errors

The cosmic bias or integral constraint:

The measurement of [FORMULA] provides the cosmic bias [FORMULA] as


(Colombi, priv. comm. 1999). This bias can be corrected in the calculation of [FORMULA] in the form of an additive factor IC. At small angular separations Eq. (24) simplifies as [FORMULA]. This negative bias is very small but becomes comparable to W when [FORMULA] approaches the angular size of the sample. The usual way of correcting for the cosmic bias is to fix a slope for the angular correlation function and to find the constant value of IC which minimizes the [FORMULA] fitting to the data. One may point out that the cosmic bias becomes non-negligible only when the errors [FORMULA] of the LS estimator given by Eq. (23) are large. The smallest scale at which [FORMULA] is [FORMULA] is [FORMULA], which corresponds to the last 2 points in all curves plotted in Fig. 12 and Fig. 13. In the interval [FORMULA] for example, [FORMULA]. However, the random errors at this scale are also significant, [FORMULA], which gives little weight to these points in the least-square fits of [FORMULA]. We therefore consider that the cosmic bias has negligible impact on our reported slope and correlation amplitude, and we chose to ignore the cosmic bias in order to avoid introducing a prior information on the slope of [FORMULA].

Misidentified stars: Because stars are uncorrelated on the sky as shown on Fig. 11, the stars fainter than [FORMULA] and [FORMULA] which are not removed from the catalogue dilute the clustering present in the galaxy correlation, i.e. decrease the amplitude of [FORMULA]. The usual way of correcting for this bias (Postman et al., 1998; Woods & Fahlman, 1997) is to apply a star dilution (multiplicative) factor [FORMULA] to the parameterized amplitude [FORMULA].


where [FORMULA] is the number of objects in the galaxy samples, and [FORMULA] is the number of stars predicted by the Bahcall model of the Galaxy (Bahcall, 1986). For [FORMULA] and [FORMULA], [FORMULA] is given by the classification efficiency of 95% of SExtractor, namely the upper value is 5% of the number of stars detected, so [FORMULA]. The values of [FORMULA] for fainter limiting magnitudes are given in Table 5.

[FIGURE] Fig. 11. Angular correlation function of stars brighter than [FORMULA]. The error bars show 1 [FORMULA] Poisson noise. The result is compatible with a random distribution.

[FIGURE] Fig. 12. Plots of Log [FORMULA] spaced by 0.5 dex (symbols; see also Table 6) and best-fit curves ([FORMULA] and [FORMULA] of Table 7) as dotted line. From top to bottom, [FORMULA] (k=5),[FORMULA] (k=4), [FORMULA] (k=3), [FORMULA] (k=2), [FORMULA] (k=1), [FORMULA] (k=0)

[FIGURE] Fig. 13. Plots of the differences of the LS [FORMULA] and the best-fit [FORMULA] (dotted lines). The curves are spaced by 1 dex for sake of clarity. Same magnitude intervals as Fig. 12.

Masking: Although the SExtractor programme is very good at avoiding false detections, it is sometimes tricked by the diffraction patterns and large wings of bright stars as would be any code using an isophotal threshold for detection. Because such false detections are strongly clustered, they increase the correlation amplitude at scales corresponding to the angular size of the false structure. To prevent the systematic patterns which could be introduced by false detections, we define two masks, one covering the bad pixels, columns and regions, and one covering all stars brighter the [FORMULA] (from the HST Guide Star Catalog). The second mask is made of four empirical components, (1) a central disk whose radius is defined by an exponential law as a function of magnitude ([FORMULA] pixels), (2) a vertical rectangle covering the bleeding streak, whose length is an exponential function of magnitude ([FORMULA] pixels), (3) and (4) inclined rectangles (at 38.5o and 51o of the vertical axis) covering the diffraction spikes, also following exponential law of the magnitude ([FORMULA] pixels). The masked regions are partially visible in Fig. 1 because only bright stars show large wings and the low density of objects does not permit one to distinguish "true" empty regions from masked regions. We apply the same two masks to the random realizations for the evaluation of LS estimator [FORMULA] in Sect. 5.1.

Photometric errors: Two kinds of photometric errors may be present: the random errors or photon noise called [FORMULA], and the residual calibration errors in the coefficients [FORMULA], [FORMULA], [FORMULA], [FORMULA] and the zero-points [FORMULA] and [FORMULA]. Note that the errors given in Table 3 called [FORMULA] and [FORMULA] include both the random and calibration errors. It is difficult but necessary to evaluate quantitatively the two kinds of errors because they have opposite qualitative effects on the angular correlation function.

The random errors can be evaluated from Table 3. It is clear that at bright magnitudes, the calibration errors dominate while random errors dominate at faint magnitudes. A random error in a galaxy apparent magnitude is equivalent to an increase of the possible volume in which that galaxy lies, i.e. it is equivalent to a convolution of the de-projected distance interval. Hence, the random error erases the clustering present in the sample. It is difficult to evaluate directly the decrease in the amplitude of [FORMULA] due to the random errors because it is impossible to disentangle it from a real variation of the spatial galaxy clustering. Nevertheless, one can follow a simple argument to estimate how random errors affect the measurement of [FORMULA]. First, assume that the galaxy number counts follow a power-law [FORMULA] so the relative error is [FORMULA]. An extreme case is to consider that all the galaxies with a magnitude error superior to the magnitude bin for which [FORMULA] is evaluated ([FORMULA]), are uncorrelated to the sample actually falling in the bin. This is an extreme case because many of the galaxies with the large error in magnitude do belong to the bin. In that case, these galaxies would have a very similar effect on the amplitude of [FORMULA] as if they were stars. So the multiplicative diluting factor [FORMULA] of the random photometric error would be,


where [FORMULA] is the magnitude error and [FORMULA] is the slope of the galaxy number count power-law. Taking the best fit slope [FORMULA] (see Fig. 8 and Table 4), a typical magnitude error of 0.15 (see Table 3) leads to a dilution factor of [FORMULA], of order of the star dilution factor [FORMULA] given in Table 5. This estimate of [FORMULA] is an indicative upper limit, and cannot be used to correct for the dilution due to random errors in the magnitude of the galaxies because the prior condition is that these galaxies are uncorrelated. This assumption might not be true, and the correction would then artificially increase the amplitude of the correlation function.


Table 4. Differential V and I-band galaxy counts (see Fig. 8).


Table 5. Star dilution correction factor [FORMULA] in V and I bands

We now evaluate the calibration error budget. Because the largest airmass difference in V is 0.267, and 0.028 in I, even a [FORMULA] error on [FORMULA] and [FORMULA] would not produce more than [FORMULA] and [FORMULA]. If we assume that the chip-to-chip error on [FORMULA] is given by the difference between the values of the synthetic sequence and the measurement of Cuillandre in Table 2 (this is certainly not the case for [FORMULA], because Cuillandre's value is too low) then in the most extreme colours [FORMULA] (only a very small fraction of our sample), the resulting magnitude difference would be [FORMULA]. The case of [FORMULA] is not clear, although it seems reasonable to assume that the error cannot be more than ten times the error [FORMULA], so [FORMULA]. The residual systematic errors in the zero-points have been evaluated in Sect. 3.2 to be [FORMULA]. Combining all these mentioned sources of calibration errors, one finds a value of 0.053 in V and an upper value of 0.094 in I.

The residual photometric errors have an opposite effect on [FORMULA]. Namely, they introduces CCD-to-CCD variations in the galaxy number counts wrongly interpreted by the correlation analysis as intrinsic clustering on a scale given by the angular size of the individual CCD chips, thus artificially increasing the amplitude of [FORMULA] on these scales. Geller et al. (1984) showed that plate-to-plate systematic variations of more than 0.05 mag introduced in the Shane-Wirtanen catalogue would produce a flattening of the correlation function and an artificial break at a scale corresponding to the plate size. For that reason, we limit our measurement of the angular correlation to [FORMULA], corresponding to the smallest dimension of the individual CCD's. We point out that the combined effect of random and zero-point errors would be to flatten artificially the slope of [FORMULA], as Geller et al. demonstrated.

Astrometric errors: Small scale astrometric errors are likely to become significant when the bin size of [FORMULA] is of the order of the error. The trivial method to avoid such contamination is to limit the analysis to bins greater than the errors, i.e. greater than [FORMULA] in our case. When one builds a mosaic survey combining overlapping patches of the sky, systematic astrometric errors may come into play: the number of objects may be artificially higher or smaller in overlapping regions just because of misidentifications. This would induce a similar effect on the amplitude of correlation to the zero-point errors at an angular scales of the order of the field size. However, a comparative analysis of the number counts of overlapping regions with other parts of the field does not show any significant bias toward a higher number of objects in the overlapping regions. We therefore consider that our astrometric errors have a negligible effect on [FORMULA], and we limit its calculation to [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001