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.
