Cosmic Error and Statistics of Large-Scale Structure

Abstract

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 {open_quotes}cosmic error,{close_quotes} 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 {ital finite} {ital volume} {ital 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 {ital edge} {ital 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 {ital discreteness} {ital effect} is due to the fact that the underlying smooth random field is sampled with finite number of objects. This is the {open_quotes}shot noise{close_quotes} error. To check the validity of our results, we measured the factorial momentsmore » of order {ital N}{le}4 in a large number of small subsamples randomly extracted from a hierarchical sample realized by Raighley-Levy random walks. The measured statistical errors are in excellent agreement with our predictions. The probability distribution of errors is increasingly skewed when the order {ital N} and/or the cell size increases. This suggests that {open_quotes}cosmic errors{close_quotes} 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. (Abstract Truncated)« less