Sampling artifact in volume weighted velocity measurement. I. Theoretical modeling

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.