Massive spectroscopic redshift surveys open a promising window to accurately measure peculiar velocity at cosmological distances through redshift space distortion (RSD). In Paper I Zhang et al. [Phys. Rev. D 87, 063526 (2013)] of this series of work, we proposed decomposing peculiar velocity into three eigenmodes (v , v S , and v B ) in order to facilitate the RSD modeling and peculiar velocity reconstruction. In the current paper we measure the dark matter RSD-related statistics of the velocity eigenmodes through a set of N-body simulations. These statistics include the velocity power spectra, correlation functions, onepoint probability distribution functions, cumulants, and the damping functions describing the Finger of God effect. We have carried out a number of tests to quantify possible numerical artifacts in these measurements and have confirmed that these numerical artifacts are under control. Our major findings are as follows: (1) The power spectrum measurement shows that these velocity components have distinctly different spatial distribution and redshift evolution, consistent with predictions in Paper I. In particular, we measure the window functionWðk; zÞ.W describes the impact of nonlinear evolution on the v -density relation. We confirm that the approximationW ¼ 1 can induce a significant systematic error of Oð10%Þ in RSD cosmology. We demonstrate thatW can be accurately described by a simple fitting formula with one or two free parameters. (2) The correlation function measurement shows that the correlation length is Oð100Þ, Oð10Þ, and Oð1Þ Mpc for v , v S , and v B , respectively. These correlation lengths determine where we can treat the velocity fields as spatially uncorrelated. Hence, they are important properties in RSD modeling.(3) The velocity probability distribution functions and cumulants quantify non-Gaussianities of the velocity fields. We confirm speculation in Paper I that v is largely Gaussian, but with non-negligible non-Gaussianity. We confirm that v B is significantly non-Gaussian. We also measure the damping functions. Despite the observed non-Gaussianities, the damping functions and hence the Finger of God effect are all well approximated as Gaussian ones at scales of interest.

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...

The mapping of dark matter clustering from real space to redshift space introduces the anisotropic property to the measured density power spectrum in redshift space, known as the redshift space distortion effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite cross correlations between the density and velocity fields, and the Finger-of-God effect due to the randomness of the peculiar velocity field. Whilst the full higher order polynomials remain unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown in this paper. The systematic due to the unknown non-linear density and velocity fields is removed by separately measuring all terms in the expansion directly using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the "one-point" FoG term being independent of the separation vector between two different points, and 2) the "correlated" FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. Using 100 realizations of simulations, we find that the Gaussian FoG function with only one scale-independent free parameter works quite well, and that our new mapping formulation accurately reproduces the observed 2-dimensional density power spectrum in redshift space at the smallest scales by far, up to k ∼ 0.2 h Mpc −1 , considering the resolution of future experiments. 98.80.Es; 98.80.Bp; 95.36.+x

Shortly after its discovery, General Relativity (GR) was applied to predict the behavior of our Universe on the largest scales, and later became the foundation of modern cosmology. Its validity has been verified on a range of scales and environments from the Solar system to merging black holes. However, experimental confirmations of GR on cosmological scales have so far lacked the accuracy one would hope for — its applications on those scales being largely based on extrapolation and its validity there sometimes questioned in the shadow of the discovery of the unexpected cosmic acceleration. Future astronomical instruments surveying the distribution and evolution of galaxies over substantial portions of the observable Universe, such as the Dark Energy Spectroscopic Instrument (DESI), will be able to measure the fingerprints of gravity and their statistical power will allow strong constraints on alternatives to GR. In this paper, based on a set of N-body simulations and mock galaxy catalogs, we study the predictions of a number of traditional and novel summary statistics beyond linear redshift distortions in two well-studied modified gravity models — chameleon f(R) gravity and a braneworld model — and the potential of testing these deviations from GR using DESI. These summary statistics employ a wide array of statistical properties of the galaxy and the underlying dark matter field, including two-point and higher-order statistics, environmental dependence, redshift space distortions and weak lensing. We find that they hold promising power for testing GR to unprecedented precision. The major future challenge is to make realistic, simulation-based mock galaxy catalogs for both GR and alternative models to fully exploit the statistic power of the DESI survey (by matching the volumes and galaxy number densities of the mocks to those in the real survey) and to better understand the impact of key systematic effects. Using these, we identify future simulation and analysis needs for gravity tests using DESI.