On the Fast Random Sampling and Other Properties of the Three Point Correlation Function in Galaxy Surveys

Abstract: In the forthcoming large volume galaxy surveys higher order statistics will provide complementary information to the usual two point statistics. Low variance estimators of the Three Point Correlation Function (3CPF) of discrete data count triangle configurations with vertices mixing data and random catalogues. Large density random catalogues are used to reduce the shot noise, which leads to a computational cost of one or two orders of magnitude more than the pure data histogram. In this paper, we explore time reductions of the isotropic 3PCF random sampling terms in periodic boxes without using random catalogues. In the first approach, based on Hamilton's construction of his famous two point estimator, we use an ad-hoc two point correlation term, while for the second procedure we construct the operators from a geometrical viewpoint, using two sides and their opening angle to describe the 3PCF triangle configurations. We map the last result to the three triangle side basis either numerically or analytically, and show that the latter approach performs best when applied to synthetic data. Moreover, we elaborate on going beyond periodic boxes, discuss other low variance n-point estimators and present useful 3PCF visualization schemes.