Cosmic Error and Statistics of Large-Scale Structure

We use a generating function approach to examine the errors on quantities related to counts in cells extracted from galaxy surveys. The measurement error, related to the finite number of sampling cells, is distinguished from the cosmic error," due to the finiteness of the survey. While the measurement error can be circumvented through the application of a proper algorithm, the cosmic error is an irrecoverable property of any survey. Using the hierarchical model and assuming locally Poisson behavior, we identify three contributions to the cosmic error: 1. The finite volume effect is proportional to the average of the two-point correlation function over the whole survey. It accounts for possible fluctuations of the density field at scales larger than the sample size. 2. The edge effect is related to the geometry of the survey. It accounts for the fact that objects near the boundary carry less statistical weight than those further away from it. 3. The discreteness effect is due to the fact that the underlying smooth random field is sampled with finite number of objects. This is the "shot noise error. To check the validity of our results, we measured the factorial moments of order N ≤ 4 in a large number of small subsamples randomly extracted from a hierarchical sample realized by Raighley-Lévy random walks. The measured statistical errors are in excellent agreement with our predictions. The probability distribution of errors is increasingly skewed when the order N and/or the cell size increases. This suggests that "cosmic errors" tend to be systematic: it is likely to underestimate the true value of the factorial moments. Our study of the various regimes showed that the errors strongly depend on the clustering of the system, i.e., on the hierarchy of underlying correlations. The Gaussian approximation is valid only in the weakly nonlinear regime, otherwise it severely underestimates the true errors. We study the concept of "number of statistically independent cells" (re)defined as the number of sampling cells required to have the measurement error of same order as the cosmic error. This number is found to depend highly on the statistical object under study and is generally quite different from the number of cells needed to cover the survey volume. In light of these findings we advocate high oversampling for measurements of counts in cells. As a preliminary application to realistic situations, we study contour plots of the cosmic error expected in typical three-dimensional galaxy catalogs.