Expansion series of the pairwise velocity generating function and its implications on redshift space distortion modeling

The pairwise velocity generating function $G$ has deep connection with both the pairwise velocity probability distribution function and modeling of redshift space distortion (RSD). Its implementation into RSD modeling is often faciliated by expansion into series of pairwise velocity moments $\langle v_{12}^n\rangle$. Motivated by the logrithmic transformation of the cosmic density field, we investigate an alternative expansion into series of pairwise velocity cumulants $\langle v_{12}^n\rangle_c$ . We numerically evaluate the convergence rate of the two expansions, with three $3072^3$ particle simulations of the CosmicGrowth N-body simulation series. (1) We find that the cumulant expansion performs significantly better, for all the halo samples and redshifts investigated. (2) For modeling RSD at $k_{\|}<0.1 h$ Mpc$^{-1}$, including only the $n=1,2$ cumulants is sufficient. (3) But for modeling RSD at $k_\parallel=0.2 h$ Mpc$^{-1}$, we need and only need the $n=1,2,3,4$ cumulants. These results provide specific requirements on RSD modeling in terms of $m$-th order statistics of the large scale strucure.