ON THE USE OF THE CONDITIONAL DENSITY AS A DESCRIPTION OF GALAXY CLUSTERING

It has been argued that the conditional density is superior to the two-point correlation function in providing a more sample-independent description of galaxy correlations and the approach to homogeneity (Pietronero). This is demonstrated here by applying both statistics to two different synthetic distributions of points: a Levy flight fractal with no intrinsic scale and a Voronoi tesselation with points distributed uniformly on the walls of cells having a well-defined mean size. In the first case, the conditional density is a well-defined power law out to the limits where it can be reliably estimated, while the two-point correlation function approaches zero on a length scale which is sample dependent giving a false indication of homogeneity. In the second case, the conditional density is a power law with the correct exponent (-1) out to the mean cell-size where it flattens correctly indicating the approach to homogeneity, while the two-point correlation function yields an incorrect estimate of the exponent (-1.8) and no obvious feature connected with the mean cell size. These synthetic distributions are also used to test two suggested methods of correcting for the boundaries of a finite sample, and it is shown that the standard method of normalizing pair counts by a Monte Carlo routine is the most accurate and unbiased. This method is then applied to the CfA catalogue where the edge-correction allows us to extend the estimate of the conditional density beyond that of previous work (Coleman et al.) to scales comparable to the depths of various volume limited subsamples. The result is that galaxy distribution apparently crosses over to homogeneity on a scale of roughly 30 Mpc and that on smaller scales the conditional density is a power law with an exponent closer to -1 (as for the Voronoi tesselation) rather than the traditional -1.8.