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

Abstract

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 δ ), an irrotational component uncorrelated with the density field (vS) and a rotational (curl) component (vB). 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) vB damps the redshift space clustering. (II) vS 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 to the Kaiser formula. Here, u ≡ kz/k, k is the amplitude of the wavevector and kz is the component along the line of sight. (III) v δ 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 vS → 0 and vB → 0, the leading order contribution ∝ (1 + fW (k)u 2 ) 2 . It differs from the Kaiser formula by a window functionW (k). Nonlinear evolution generically drivesW (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 relatively large scale k = 0.1h/Mpc. (IV) The velocity decomposition reveals the three origins of the finger of God (FOG) effect and suggests 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 correction of order δ 3 can be taken into account without introducing extra model uncertainties. Here δ 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 δ 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. Both can automatically and significantly alleviate the galaxy bias problem and, in the limit of a deterministic galaxy bias, completely overcome it. Paper I of this series of works lays out the methodology. Companion papers [1] will extensively evaluate its performance against N-body sim...

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

“…Comprehensive study on related statistics and tests against simulations will be presented in paper II. This is along our recent effort to understand peculiar velocity and redshift space distortion [1, 9,10]. When measur-ing the halo velocity statistics [10], we find significant sampling artifact.…”

Sampling artifact in volume weighted velocity measurement. II. Detection in simulations and comparison with theoretical modeling

“…This will then allow us to measure the real halo velocity bias, free of otherwise severe systematic error from the sampling artifact. We will present such measurements in [5], which belongs to our ongoing efforts toward understanding the velocity field, redshift shift space distortion, and velocity reconstruction in spectroscopic redshift surveys [12,24]. The current paper focuses on the sampling artifact in the gradient part of the velocity field.…”

Helmholtz decomposition of the Lagrangian displacement

“…This kind of decomposition has been widely used in physics, for example, in the decomposition of the electric field in electromagnetism [29], and the cosmological perturbation theory [30]. Recently, it has been applied to redshift space distortion as different components correspond to different physical origins [31,32]. The scalar and vector potentials can be solved 1 The uniqueness of the decomposition generally depends on the boundary conditions.…”