Modelling redshift space distortions in hierarchical cosmologies

TL;DR

The paper evaluates redshift-space distortions in ΛCDM and dynamical quintessence cosmologies using large-volume N-body simulations, showing that linear Kaiser predictions fail even on surprisingly large scales. It develops a non-linear framework based on P_{δδ}, P_{δθ}, and P_{θθ}, and demonstrates that incorporating the density–velocity relation yields accurate, cosmology-independent predictions up to k ≈ 0.2–0.3 h Mpc^{-1}. A practical redshift-dependent mapping relates P_{δθ} and P_{θθ} to the non-linear matter power spectrum using the linear growth factor, achieving better than ~5–10% accuracy across models. These results enhance the extraction of growth-rate information from future galaxy surveys by providing robust, parameter-efficient redshift-space models.

Abstract

The anisotropy of clustering in redshift space provides a direct measure of the growth rate of large scale structure in the Universe. Future galaxy redshift surveys will make high precision measurements of these distortions, and will potentially allow us to distinguish between different scenarios for the accelerating expansion of the Universe. Accurate predictions are needed in order to distinguish between competing cosmological models. We study the distortions in the redshift space power spectrum in $Λ$CDM and quintessence dark energy models, using large volume N-body simulations, and test predictions for the form of the redshift space distortions. We find that the linear perturbation theory prediction by Kaiser (1987) is a poor fit to the measured distortions, even on surprisingly large scales $k \ge 0.05 h$Mpc$^{-1}$. An improved model for the redshift space power spectrum, including the non-linear velocity divergence power spectrum, is presented and agrees with the power spectra measured from the simulations up to $k \sim 0.2 h$Mpc$^{-1}$. We have found a density-velocity relation which is cosmology independent and which relates the non-linear velocity divergence spectrum to the non-linear matter power spectrum. We provide a formula which generates the non-linear velocity divergence $P(k)$ at any redshift, using only the non-linear matter power spectrum and the linear growth factor at the desired redshift. This formula is accurate to better than 5% on scales $k<0.2 h $Mpc$^{-1}$ for all the cosmological models discussed in this paper. Our results will extend the statistical power of future galaxy surveys.

Figures (9)

• Figure 1: Left panel: The linear growth factor divided by the scale factor as a function of redshift for the SUGRA and CNR quintessence models and $\Lambda$CDM, as indicated by the key. Right panel: The linear growth rate, $f = {\rm{d}} {\rm{ln}} D/{\rm{d ln}} a$, for the two dark energy models and $\Lambda$CDM as a function of redshift. In both the left and right main panels, solid lines represent the exact solution for the linear growth factor and growth rate and dashed lines show the fitting formula given in Eq. \ref{['linder']}. Note in the right main panel the $\Lambda$CDM grey dashed line has been omitted for clarity. The lower left hand panel shows the formula for $D(a)/a$ given by Linder:2005in divided by the exact solution as a function of redshift. The ratio of the formula in Eq. \ref{['linder']} for the growth rate, $f$, to the exact solution is shown in the lower right hand panel. Also in the lower right panel the dotted lines show the ratio of the fitting formula $f= \Omega^{0.6}_{\rm m}$ to the exact solution for each of the dark energy models plotted as a function of redshift.
• Figure 2: Left panel: The ratio of the monopole redshift power spectra and real space power spectra measured from the $\Lambda$CDM simulation at $z=0$ and $z=1$ are plotted as blue lines. The error bars plotted represent the scatter between the different power spectra from four $\Lambda$CDM simulations set up with different realisations of the density field with the distortions imposed along either the $x, y$ or $z$ axis and averaged. The power spectra $P(k,\mu=k_x/k)$, $P(k,\mu=k_y/k)$ and $P(k,\mu=k_z/k)$ measured from one simulation are plotted as the cyan, purple and red dashed lines respectively. Right panel: The ratio of the quadrupole to monopole moment of the redshift space power spectrum measured from the simulations at $z=0$ and $z=1$ in $\Lambda$CDM are plotted in blue. It was not possible to accurately measure the quadrupole to monopole power in the first bin, so this point has not been plotted in the right hand panel. Note for wavenumbers $k>0.1h$Mpc$^{-1}$, only every fifth error bar is plotted for clarity. The Kaiser formula, given by Eq. \ref{['mr']}, is plotted as a blue dotted line. The error bars were obtained as described for the left-hand panel.
• Figure 3: Left panel: The ratio of the non-linear power spectra, $P_{\delta \delta}$, $P_{\delta \theta}$ and $P_{\theta \theta}$ for $\Lambda$CDM measured from the simulation at $z=0$, divided by the corresponding power spectrum measured from the simulation at $z=5$, scaled using the square of the ratio of the linear growth factor at $z=5$ and $z=0$. The non-linear matter power spectrum is plotted as a grey dot-dashed line, the non-linear velocity divergence auto power spectrum $P_{\theta \theta}$ is plotted as a blue solid line and the non-linear cross power spectrum, $P_{\delta \theta}$, is plotted as a green dashed line. Right panel: The ratio of the non-linear power spectra, $P_{\delta \delta}$, $P_{\delta \theta}$ and $P_{\theta \theta}$, to the linear theory matter $P(k)$ in $\Lambda$CDM measured from the simulation at $z=0$. All power spectra have been divided by the linear theory matter power spectrum measured from the simulation at $z=5$, scaled using the square of the ratio of the linear growth factor at $z=5$ and $z=0$. In both panels the error bars represent the scatter over eight $\Lambda$CDM realisations after imposing the peculiar velocity distortion along each cartesian axis in turn.
• Figure 4: Left panel: The ratio of the non-linear power spectra, $P_{\delta \delta}$, $P_{\delta \theta}$ and $P_{\theta \theta}$, to the linear theory $P(k)$ in $\Lambda$CDM measured from one realisation of the matter density and velocity fields at $z=0$. All power spectra have been divided by the linear theory matter power spectrum measured from the simulation at $z=5$, scaled using the square of the ratio of the linear growth factor at $z=5$ and $z=0$. Right panel: Similar to that in the left panel but for the SUGRA quintessence model. The lines are the same as used in the left hand panel.
• Figure 5: A comparison of the impact of the FFT grid dimension on power spectrum estimation. The plots show the ratio of the non-linear power spectra, $P_{\theta \theta}$ (upper panel) and $P_{\delta \theta}$ (lower panel), to the linear theory matter power spectrum measured from the simulations in $\Lambda$CDM, using different FFT grid sizes. From bottom to top in each panel the lines show the ratios for grid sizes $N_{\tiny\hbox{FFT}} =128$ (purple), $N_{\tiny\hbox{FFT}} =256$ (blue), $N_{\tiny\hbox{FFT}} =350$ (red) and $N_{\tiny\hbox{FFT}} =375$ (green).