We study the connections between the pairwise velocity moment generating function G(k _∥, r) and redshift space distortion (RSD) modeling. Here k _∥ is the Fourier wavevector parallel to the line of sight, and r is the pair separation vector. We demonstrate its usage by two examples. (1) Besides the known relations between G and the RSD power spectrum (and the correlation function), we propose a new RSD statistics P^s (k _∥,r _⊥) whose connection to G is convenient to evaluate numerically. (2) We then develop a fast method to numerically evaluate G, and apply it to a high resolution N-body simulation. We find that G (ln G) shows complicated dependence on k _∥ beyond the linear and quadratic dependencies. This not only shows inaccuracy in some existing models and identifies sources of inaccuracy but also provides possible ways of improving the RSD modeling. Consequently, more comprehensive investigations on G are needed to fully explore the usage of G in RSD modeling.

我们研究了成对速度矩生成函数G ( k _∥ , r ) 与红移空间畸变 (RSD) 建模之间的联系。这里k∥ 是平行于视线的傅里叶波向量，_r是对分离向量。我们通过两个例子来演示它的用法。(1) 除了G与 RSD 功率谱（以及相关函数）之间的已知关系外，我们提出了一个新的 RSD 统计P ^s ( k _∥ , r _⊥ )，它与G的连接便于数值评估。(2) 然后我们开发了一种快速数值评估G的方法，并将其应用于高分辨率 N 体模拟。我们发现G (ln G )在线性和二次依赖之外表现出对k _∥的复杂依赖。这不仅显示了一些现有模型的不准确性并确定了不准确性的来源，而且还提供了改进 RSD 建模的可能方法。因此，需要对G进行更全面的研究，以充分探索G在 RSD 建模中的用途。