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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., 19 pages, 4 figures

We present a simple closed-form bandwidth selection method to obtain the optimal bin size for estimating the two-point correlation function (2PCF). To demonstrate its effectiveness, we apply this procedure in a numerical study using a theoretical point process model to 10 mock Sloan Digital Sky Survey (SDSS) catalogs with the same cosmology, and to a real SDSS catalog. With a point process model, the real 2PCF is known and ratios of the mean square errors of estimates can be computed. The standard deviation of 2PCF estimates, obtained from the 10 mock catalogs, provides information about the uncertainty of estimates. With the SDSS catalog, we use subsampling to assess uncertainty. In each case, we find that our procedure performs better than the commonly used procedure of equal bandwidths on the log-scale.

We examine the three-dimensional clustering of C IV absorption-line systems using an extensive catalog of QSO heavy-element absorbers drawn from the literature. We measure clustering by a volume-weighted integral of the correlation function called the reduced second-moment measure, and include information from both along and across QSO lines of sight, thus enabling a full determination of the three-dimensional clustering of absorbers, as well as a comparison of along- and across-line-of-sight clustering properties. Here we present the three-dimensional reduced second-moment estimator for a three-dimensional point process probed by one-dimensional lines of sight, and apply our algorithm to a sample of 345 C IV absorbers with median redshift z = 2.2, drawn from the spectra of 276 QSOs. We confirm the existence of significant clustering on comoving scales of up to 100 h-1 Mpc (q0 = 0.5), and find that the additional across-line-of-sight information strengthens the evidence for clustering on scales from 100 to 150 h-1 Mpc. There is no evidence of absorber clustering along or across lines of sight for scales from 150 to 1000 h-1 Mpc. We show that with a 300 times larger catalog, such as that to be compiled by the Sloan Digital Sky Survey (100,000 QSOs), use of the full three-dimensional estimator and across-line-of-sight information will substantially increase clustering sensitivity. We find that standard errors are reduced by a factor of 2-20 on scales of 30-200 h-1 Mpc, in addition to the factor of (300)1/2 reduction from the larger sample size, effectively increasing the sample size by an extra factor of 4-400 at large distances.