## 5. Analysis## 5.1. Estimation ofThe 2-point angular correlation function 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 defined by Landy & Szalay (1993) (hereafter LS), which has the advantage of reduced edge effects and smallest possible variance: where then Eq. (8) can be re-written as is the probability of finding
two randomly placed galaxies separated by a distance
, where is the average number of unique triplets. is necessary to evaluate the random errors (cf Sect. 5.3). ## 5.2. Modeling of , andThe canonical parameterizations of the two-point spatial correlation function (Phillipps et al., 1978) and of the angular correlation function (Peebles, 1980) are where Given Eq. (14) and the galaxy redshift distribution , one can relate and : Here is the angular diameter
distance, is the comoving volume
element, both given for three cosmologies in the appendix of Cole et
al. (1992). is the luminosity
function, whose definition might be dependent on different spectral
type ( is rewritten as ). ## 5.3. Estimation of errorsIn the past few years, considerable theoretical efforts have been devoted to the calculation of errors in the estimation of . 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 errorsUntil recently, a Gaussian approximation for the distribution of the galaxies was assumed to calculate the errors on . 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 . Bernstein derives a formal solution to the random errors for the case of a hierarchical clustering universe in the limits (number of galaxies in the sample), , and angular size of the sample: where the parameters and
are measured by Gaztañaga
(1994). They are related to the hierarchical amplitudes
and
(, where
are the n-point angular correlation
functions), by and
(Gazta"naga, 1994).
and
have been derived by
Gaztañaga (1994) and Roche & Eales (1999) from the APM
catalogue (Maddox et al., 1990a):
and at
, the scale at which the
angular-correlation function is well measured in the APM catalogue
(Maddox et al., 1990b); therefore
and .
( ## 5.3.2. Systematic errors
The measurement of provides the cosmic bias as (Colombi, priv. comm. 1999). This bias can be corrected in the
calculation of in the form of an
additive factor
where is the number of objects in the galaxy samples, and is the number of stars predicted by the Bahcall model of the Galaxy (Bahcall, 1986). For and , is given by the classification efficiency of 95% of SExtractor, namely the upper value is 5% of the number of stars detected, so . The values of for fainter limiting magnitudes are given in Table 5.
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 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 . First, assume that the galaxy number counts follow a power-law so the relative error is . An extreme case is to consider that all the galaxies with a magnitude error superior to the magnitude bin for which is evaluated (), 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 as if they were stars. So the multiplicative diluting factor of the random photometric error would be, where is the magnitude error and is the slope of the galaxy number count power-law. Taking the best fit slope (see Fig. 8 and Table 4), a typical magnitude error of 0.15 (see Table 3) leads to a dilution factor of , of order of the star dilution factor given in Table 5. This estimate of 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.
We now evaluate the calibration error budget. Because the largest
airmass difference in The residual photometric errors have an opposite effect on . 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 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 , 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 , as Geller et al. demonstrated.
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |