Sampling Artifact in Volume Weighted Velocity Measurement.--- I. Theoretical Modelling

Cosmology based on large scale peculiar velocity preferes volume weighted velocity statistics. However, measuring the volume weighted velocity statistics from inhomogeneously distributed galaxies (simulation particles/halos) suffer from an inevitable and significant sampling artifact. We study this sampling artifact in the velocity power spectrum measured by the nearest-particle (NP) velocity assignment method(Zheng et al. 2013, PRD). We derive the analytical expression of leading and higher order terms. We find that the sampling artifact suppresses the $z=0$ E-mode velocity power spectrum by $\sim 10\%$ at $k=0.1h/$Mpc , for samples with number density $10^{-3}({\rm Mpc}/h)^{-3}$. This suppression becomes larger for larger $k$ and for sparser samples. We argue that, this source of systematic errors in peculiar velocity cosmology, albeit severe, can be self-calibrated in the framework of our theoretical modelling. We also work out the sampling artifact in the density-velocity cross power spectrum measurement. More robust evaluation of related statistics through simulations will be presented in a companion paper (Zheng, Zhang & Jing, 2015, PRD). We also argue that similar sampling artifact exists in other velocity assignment methods and hence must be carefully corrected to avoid systematic bias in peculiar velocity cosmology.