Estimating Third-Order Moments for an Absorber Catalog

Abstract

Thanks to the recent availability of large surveys, there has been renewed interest in third-order correlation statistics. Measures of third-order clustering are sensitive to the structure of filaments and voids in the universe and are useful for studying large-scale structure. Thus, statistics of these third-order measures can be used to test and constrain parameters in cosmological models. Third-order measures such as the three-point correlation function are now commonly estimated for galaxy surveys. Studies of third-order clustering of absorption systems will complement these analyses. We define a statistic, which we denote K, that measures third-order clustering of a data set of point observations and focus on estimating this statistic for an absorber catalog. The statistic K can be considered a third-order version of the second-order Ripley K-function and allows one to study the abundance of various configurations of point triplets. In particular, configurations consisting of point triplets that lie close to a straight line can be examined. Studying third-order clustering of absorbers requires consideration of the absorbers as a three-dimensional process, observed on QSO lines of sight that extend radially in three-dimensional space from Earth. Since most of this three-dimensional space is not probed by the lines of sight, edge corrections become important. We use an analytical form of edge correction weights and construct an estimator of the statistic K for use with an absorber catalog. We show that with these weights, ratio-unbiased estimates of K can be obtained. Results from a simulation study also verify unbiasedness and provide information on the decrease of standard errors with increasing number of lines of sight.