Peculiar velocity decomposition, redshift space distortion, and velocity reconstruction in redshift surveys: The methodology

Massive spectroscopic surveys will measure the redshift space distortion (RSD) induced by galaxy peculiar velocity to unprecedented accuracy and open a new era of precision RSD cosmology. 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 eigencomponents: an irrotational component completely correlated with the underlying density field (v_), an irrotational component uncorrelated with the density field (vₒ), and a rotational (curl) component (v₁). The three components have different origins, different scale dependences, and different impacts on RSD. (2) This decomposition has the potential to simplify and improve the RSD modeling. (i) v₁ damps the redshift space clustering. (ii) vₒ causes both damping and enhancement to the redshift space power spectrum P^s (k, u). Nevertheless, the leading order contribution to the enhancement has a u^4 directional dependence, distinctively different from the Kaiser formula. Here, ukₙ/k, k is the amplitude of the wave vector, and kₙ is the component along the line of sight. (iii) v_ is of the greatest importance for the RSD cosmology. We find that the induced redshift clustering shows a number of important deviations from the usual Kaiser formula. Even in the limit of vₒ0 and v₁0, the leading order contribution (1+fW (k) u^2) ^2. It differs from the Kaiser formula by a window function W (k). Nonlinear evolution generically drives W (k) 1. We hence identify a significant systematical error causing underestimation of the structure growth parameter f by as much as O (10%) even at a relatively large scale k=0. 1h/Mpc. (iv) The velocity decomposition reveals the three origins of the ``finger-of-God'' (FOG) effect and suggests how to simplify and improve the modeling of FOG by treating the three components separately. (v) We derive a new formula for the redshift space power spectrum. Under the velocity decomposition scheme, all high order Gaussian corrections and non-Gaussian corrections of order ^3 can be taken into account without introducing extra model uncertainties. Here is the nonlinear overdensity. (3) The velocity decomposition clarifies issues in peculiar velocity reconstruction through 3D galaxy distribution. We discuss two possible ways to carry out the 3D v_ reconstruction. Both use the otherwise troublesome RSD in velocity reconstruction as a valuable source of information. Both have the advantage of rendering the reconstruction of a stochastic 3D field into the reconstruction of a deterministic window function W^s (k, u) of limited degrees of freedom. Both can automatically and significantly alleviate the galaxy bias problem and, in the limit of a deterministic galaxy bias, completely overcome it. Part 1 of this series of works lays out the methodology. Companion papers Y. Zheng et al. (in preparation). will extensively evaluate its performance against N-body simulations.