Boundary Corrections in Fractal Analysis of Galaxy Surveys

The analysis of redshift surveys with fractal tools requires one to apply some form of statistical correction for galaxies lying near the geometric boundary of the sample. In this paper we compare three different methods of performing such a correction upon estimates of the correlation integral in order to assess the extent to which estimates may be biased by boundary terms. We apply the corrections illustrative examples, including a simple fractal set (L\'{e}vy flight), a random $\beta$-model, and a subset of the CfA2 Southern Cap survey. This study shows that the new ``angular'' correction method we present is more generally applicable than the other methods used to date: the conventional ``capacity'' correction imposes a bias towards homogeneity, and the ``deflation'' method discards large-scale information, and consequently reduces the statistical usefulness of data sets. The ``angular'' correction method is effective at recovering true fractal dimensions, although the extent to which boundary corrections are important depends on the form of fractal distribution assumed as well as the details of the survey geometry. We also show that the CfA2 Southern sample does not show any real evidence of a transition to homogeneity. We then revisit the IRAS PSCz survey and ``mock'' PSCz catalogues made using N-body simulations of two different cosmologies. The results we obtain from the PSCz survey are not significantly affected by the form of boundary correction used, confirming that the transition from fractal to homogeneous behaviour reported by Pan & Coles (2000) is real.