The Poisson noise in modeling the redshift-space distortion at large scales

Abstract

We investigate the errors in modeling the redshift-space distortion (RSD) effect at large linear scales, using data from the Millennium simulation. While standard theoretical templates, such as the Kaiser formula and the TNS method, could precisely model RSD for individual large-scale modes, we find that for tracers with number densities lower than $\sim10^{-3}({\rm Mpc}/h)^{-3}$, there is a few-percent level bias in the predicted power spectrum. This error arises due to the amplification of intrinsic Poisson noise during RSD modeling from real-space power spectrum. This amplified noise can be analytically expressed as $1 + \epsilon/[{\bar{n}P}({1+\epsilon})]$, with $\epsilon=2\beta/3+\beta^2/5$, where $P$ denotes the real-space tracer power spectrum and $\beta \equiv f/b$. Specifically, for halos with a number density of around $5\times10^{-4}({\rm Mpc}/h)^{-3}$, this phenomenon results in an additional systematic error of 2.5\%. Our result suggests that caution is necessary when directly modeling redshift-space distortions (RSD) using real-space power spectra of tracers obtained from simulations or actual surveys. This caution is particularly pertinent in scenarios where emulators trained on simulation data forecast the real-space tracer power spectrum, as well as in baryon acoustic oscillation (BAO) reconstruction using galaxy samples, for which we estimate that shot noise could introduce random errors of about one-third in the displacement field, potentially diminishing the effectiveness of the BAO peak sharpening.