Redshift-space distortions, pairwise velocities, and nonlinearities

We derive the exact relationship, including all nonlinearities, between real-space and redshift-space two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity probability distribution function is strongly non-Gaussian at all scales and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive. We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well.

Figure 1

The parallel to the line of sight pairwise velocity PDF at redshift $z=0$ for pairs separated by distance $r$, measured in the $N$-body simulations. In the bottom left panel, the discontinuous at the origin PDF (thin solid line) corresponds to that given by the dispersion model, Eq. (19) (ignoring the delta function at the origin). In the bottom right panel, the narrow distribution (thin solid line) corresponds to the prediction of linear dynamics, Eq. (44).

Figure 2

Same as Fig. 1 but for pairs perpendicular to the line of sight. The thin solid lines in the bottom panels are as in Fig. 1, the prediction of the dispersion model (left) and linear dynamics (right).

Figure 3

Moments of pairwise velocities parallel to the line connecting the pair as a function of scale. Top panel: pairwise dispersion ${\sigma }_{12}$ (squares) as a function of scale, its connected piece ${\sigma }_{12}^c$ (solid line), and the mean infall $v_{12}$ (triangles). The dashed line denotes the predicted ${\sigma }_{12}$ in linear dynamics. Middle panel: dimensionless measure of infall ($|v_{12}|/{\sigma }_{12}^c$, triangles) compared to the skewness of the pairwise velocity PDF (squares); the skewness dominates at most scales. Bottom panel: kurtosis as a function scale, note that it does not vanish at large scales and $s_4>1$ at all scales; the pairwise velocity PDF is strongly non-Gaussian at all scales, see Fig. 1. For reference, an exponential distribution has $s_3=0$ and $s_4=3$.

Figure 4

Contours of ${\xi }_s(s_{\parallel },s_{\perp })$ for the Exact Gaussian result (solid line) and the Kaiser limit (dashed line), for a flat $\Lambda$CDM cosmological model (${\Omega }_m=0.26$, ${\sigma }_8=0.9$, ${\Omega }_b=0.04$, $h=0.7$) with linear bias $b_1=1$.

Figure 5

Same as Fig. 4 but at small scales. Note the dispersion effect: a Gaussian random field does show “fingers of God” even though the velocity dispersion decreases monotonically towards small scales (see dashed line in Fig. 3).

Figure 6

Nonlinear corrections to the density-density (top), density-velocity (middle), and velocity-velocity (bottom) power spectra as a function of scale. The symbols denote measurements in the very large simulation dark matter simulations; solid lines denote one-loop perturbation theory.

Figure 7

Contours of the redshift-space power spectrum at $z=0$. The solid lines correspond to the $N$-body simulation results, dashed lines denote the Kaiser formula, and dotted lines show the simplified ansatz of Eq. (71).

Figure 8

Top panel: Ratio of the monopole (solid line) and quadrupole (dashed line) to the predictions of the Kaiser formula for redshit-space power given by Eq. (71). Symbols show the same quantities in the numerical simulations. Bottom panel: recovery of the real-space power spectrum from redshift-space multipoles according to Eq. (73) for the model in Eq. (71) (solid line) and $N$-body simulations (symbols).