Peculiar Velocity Decomposition, Redshift Space Distortion and Velocity
  Reconstruction in Redshift Surveys - I. The Methodology
  Massive spectroscopic surveys will measure the redshift space distortion
(RSD) induced by galaxy peculiar velocity to unprecedented accuracy. We develop
a new method to improve the RSD modeling and to carry out robust reconstruction
of the 3D large scale peculiar velocity through galaxy redshift surveys, in
light of RSD. (1) We propose a mathematically unique and physically motivated
decomposition of peculiar velocity into three eigen-components of different
origins, different scale dependences and different impacts on RSD. (2) This
decomposition has the potential to simplify and improve the RSD modeling. We
derive a new formula for the redshift space power spectrum. Under the velocity
decomposition scheme, all high order Gaussian corrections and non-Gaussian
correction of order $\delta^3$ can be taken into account without introducing
extra model uncertainties. We find that the leading order contribution $\propto
(1+f\tilde{W}(k)u^2)^2$ differs from the Kaiser formula by a function
$\tilde{W}(k)\leq 1$. We hence identify a significant systematic error causing
underestimation of the structure growth parameter $f$ by as much as $O(10%)$
even at relatively large scale $k=0.1h/$Mpc. The velocity decomposition also
reveals the three origins of the finger of God (FOG) effect. (3) The velocity
decomposition clarifies issues in peculiar velocity reconstruction through 3D
galaxy distribution. We discuss two possible ways to carry out the velocity
reconstruction. Both use the otherwise troublesome RSD in velocity
reconstruction as a valuable source of information. Both have the advantage to
render the reconstruction of a stochastic 3D field into the reconstruction of a
deterministic window function $W^s(k,u)$ of limited degrees of
freedom...[abridged]
