Distribution function approach to redshift space distortions. Part II: N-body simulations

TL;DR

This paper assesses a distribution-function approach to redshift-space distortions by expanding the redshift-space power spectrum $P^{ss}(k)$ in powers of μ, the cosine of the line-of-sight angle, and by decomposing angular dependence via helicity. Using large ΛCDM N-body simulations, it shows that a finite μ^2 expansion converges on large scales, with higher-order terms becoming significant on smaller scales and FoG effects driving strong deviations; resumming these terms into FoG kernels, approximated by Lorentzian forms with scale- and angle-dependent dispersions, extends reliable predictions to quasi-nonlinear regimes. Three separate FoG kernels corresponding to the lowest three contributing terms are needed to reproduce the monopole, quadrupole, and hexadecapole with percent-level accuracy up to about $k\sim0.4\,h\,{ m Mpc^{-1}}$ at $z=0$, and up to higher $k$ at higher redshift. The study also compares mass-weighted versus volume-weighted velocity statistics, finding mass weighting more natural and practically measurable in sparse samples. Overall, the μ^2-based expansion provides controlled convergence and a viable path to modeling RSD, with FoG resummation essential for extending accuracy into the nonlinear regime; the framework is readily extendable to halos and galaxies in future work.

Abstract

Measurement of redshift-space distortions (RSD) offers an attractive method to directly probe the cosmic growth history of density perturbations. A distribution function approach where RSD can be written as a sum over density weighted velocity moment correlators has recently been developed. We use Nbody simulations to investigate the individual contributions and convergence of this expansion for dark matter. If the series is expanded as a function of powers of mu, cosine of the angle between the Fourier mode and line of sight, there are a finite number of terms contributing at each order. We present these terms and investigate their contribution to the total as a function of wavevector k. For mu^2 the correlation between density and momentum dominates on large scales. Higher order corrections, which act as a Finger-of-God (FoG) term, contribute 1% at k~0.015h/Mpc, 10% at k~0.05h/Mpc at z=0, while for k>0.15h/Mpc they dominate and make the total negative. These higher order terms are dominated by density-energy density correlations which contribute negatively to the power, while the contribution from vorticity part of momentum density auto-correlation is an order of magnitude lower. For mu^4 term the dominant term on large scales is the scalar part of momentum density auto-correlation, while higher order terms dominate for k>0.15h/Mpc. For mu^6 and mu^8 we find it has very little power for k<0.15h/Mpc. We also compare the expansion to the full 2D P^ss(k,mu) as well as to their multipoles. For these statistics an infinite number of terms contribute and we find that the expansion achieves percent level accuracy for kmu<0.15h/Mpc at 6th order, but breaks down on smaller scales because the series is no longer perturbative. We explore resummation of the terms into FoG kernels, which extend the convergence up to a factor of 2 in scale. We find that the FoG kernels are approximately Lorentzian.

Figures (9)

• Figure 1: Power spectra measured in redshift space $P^{ss}(k,\mu)$ and individual contributions to it from the terms of the moments expansion up to 4-th order at $z=0$ (left) and $z=2$ (right). The width of $\mu$ bin is 0.2 centered around the values shown in each panel. The solid and dashed lines show the positive values, while the dotted lines the negative values.
• Figure 2: Upper panels: we show monopole moments of power spectrum in redshfit space $P^{ss}$. The vertical axis is multiplied by $k$ to clearly illustrate departures from a reference power spectrum. The green, blue, yellow, magenta and cyan lines respectively show our model prediction up to nonlinear Kaiser, 2nd, 3rd, 4th and 6th order corrections, measured from the simulations. The black lines are linear theory prediction. The black points with errorbars show the reference power spectrum. The red lines show our FoG model (section \ref{['sec:fog_resum']}). Lower panels: error between the summed power spectrum and the reference spectrum. The meaning of the color of each line is the same as that of the upper panels. For reference errorbars are shown for the result of our FoG model.
• Figure 3: Redshift-space power spectrum divided by the no-wiggle approximation Eisenstein:1998, monopole (left) and quadrupole (right). The green, blue and cyan lines show our model prediction up to nonlinear Kaiser, 2nd and 6th order corrections. The dotted black lines are linear theory prediction, while the solid black lines the reference power spectrum. The red lines show our FoG model. The negative values of the quadrupole spectrum on small scales are shown as the dotted lines.
• Figure 4: Same as figure \ref{['fig:pkl_mono']}, but for the quadrupole. The dashed lines at the top panels show positive values while the dotted lines show negative values.
• Figure 5: Same as figure \ref{['fig:pkl_mono']}, but for the hexadecapoles. We adopt the different bin sizes in logarithmic spacing at $k\leq 0.1{\hbox{$~h{\rm ~Mpc}^{-1}$}}$ and $k\geq 0.1{\hbox{$~h{\rm ~Mpc}^{-1}$}}$. Artificial cuts are put for the plots of the hexadecapoles at low $k$ because of large sampling variance.