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

TL;DR

This paper introduces an improved series expansion of the pairwise velocity generating function using cumulants, demonstrating better convergence and practical modeling implications for redshift space distortions in cosmology.

Contribution

It proposes and evaluates an alternative cumulant-based expansion for the pairwise velocity generating function, showing its superior convergence over traditional moment expansions.

Findings

Cumulant expansion outperforms moment expansion in convergence rate.

Including up to the 4th cumulant suffices for accurate RSD modeling at certain scales.

Results inform the order of statistics needed for precise large-scale structure modeling.

Abstract

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.