Redshift Space Distortion of the 21cm Background from the Epoch of
  Reionization I: Methodology Re-examined
  The peculiar velocity of the intergalactic gas responsible for the cosmic
21cm background from the epoch of reionization and beyond introduces an
anisotropy in the three-dimensional power spectrum of brightness temperature
fluctuations. Measurement of this anisotropy by future 21cm surveys is a
promising tool for separating cosmology from 21cm astrophysics. However,
previous attempts to model the signal have often neglected peculiar velocity or
only approximated it crudely. This paper re-examines the effects of peculiar
velocity on the 21cm signal in detail, improving upon past treatment and
addressing several issues for the first time. (1) We show that properly
accounting for finite optical depth eliminates the unphysical divergence of
21cm brightness temperature in overdense regions of the IGM found by previous
work that employed the usual optically-thin approximation. (2) The
approximation made previously to circumvent the diverging brightness
temperature problem by capping velocity gradient can misestimate the power
spectrum on all scales. (3) The observed power spectrum in redshift-space
remains finite even in the optically-thin approximation if one properly
accounts for the redshift-space distortion. However, results that take full
account of finite optical depth show that this approximation is only accurate
in the limit of high spin temperature. (4) The linear theory for redshift-space
distortion results in ~30% error in the observationally relevant wavenumber
range, at the 50% ionized epoch. (5) We describe and test two numerical schemes
to calculate the 21cm signal from reionization simulations to incorporate
peculiar velocity effects in the optically-thin approximation accurately. One
is particle-based, the other grid-based, and while the former is most accurate,
we demonstrate that the latter is computationally more efficient and can
achieve sufficient accuracy. [Abridged]
