Mapping Growth and Gravity with Robust Redshift Space Distortions

TL;DR

This work systematically tests redshift-space distortion (RSD) modeling against large-volume N-body simulations to quantify biases in growth-rate measurements and gravity tests. By comparing Kaiser, quasi-linear (Scoccimarro), and higher-order perturbation theories (SPT, LPT, Taruya$^{++}$) across redshifts $z=0$, 0.5, 1 and up to $k_{\max}=0.2\,h\mathrm{Mpc}^{-1}$, it shows that many common models bias the growth rate $f$ by more than $1\sigma$ at surprisingly large scales due to incorrect angular dependence and damping. The study introduces a scale- and angle-dependent damping function $F(k,\mu)$ (fit by $F(k,\mu)=\dfrac{A}{1+B k^2 \mu^2}+C k^2 \mu^2$) that can reproduce the redshift-space power spectrum to percent accuracy, enabling robust modeling to higher $k$. However, constraints on the gravitational growth index $\gamma$ remain highly sensitive to RSD modeling; extending analyses to $k_{\max}=0.2\,h\mathrm{Mpc}^{-1}$ yields biased $\gamma$ and other parameters, underscoring the need for improved modeling and possible combination with other probes for credible tests of gravity.

Abstract

Redshift space distortions caused by galaxy peculiar velocities provide a window onto the growth rate of large scale structure and a method for testing general relativity. We investigate through a comparison of N-body simulations to various extensions of perturbation theory beyond the linear regime, the robustness of cosmological parameter extraction, including the gravitational growth index, γ. We find that the Kaiser formula and some perturbation theory approaches bias the growth rate by 1-sigma or more relative to the fiducial at scales as large as k > 0.07 h/Mpc. This bias propagates to estimates of the gravitational growth index as well as Ω_m and the equation of state parameter and presents a significant challenge to modelling redshift space distortions. We also determine an accurate fitting function for a combination of line of sight damping and higher order angular dependence that allows robust modelling of the redshift space power spectrum to substantially higher k.

Figures (10)

• Figure 1: Constraints on the growth rate using the various Kaiser-type models of the redshift space power spectrum evaluated at $z=0$ (top), $z=0.5$ (middle), and $z=1$ (bottom), with cutoffs at k$_{max} = 0.07, 0.1, 0.2\,h$/Mpc. All the models are biased; the horizontal lines show the true value of $f$ at each redshift. The fitted parameters are $\{f, b\}$, and $\sigma_v$ in the case of empirical damping. Note that some points have been offset by a small amount in k$_{max}$ for clarity.
• Figure 2: $P^s(k,\mu)$ for $z=0,0.5,1$ calculated using the Kaiser limit without damping (dashed purple), the streaming model (solid green), and the streaming model using the non-linear matter power spectrum (dotted blue), compared to measured redshift space power spectra from N-body simulations (black). The numbers on the black curves indicate the contour levels in $\log{P^s(k,\mu)}$. The hyperbolic red contours show constant $k\mu=0.05\,h$/Mpc. FoG effects are manifested in the amplification of the black contours at large $k$ perpendicular to the line of sight ($\mu\approx0$).
• Figure 3: As in Fig. \ref{['fig:kaiser_f_vs_kmax']} but using the Scoccimarro ansatz with the real space non-linear power spectra evaluated using SPT. We consider two types of damping in these fits, exponential damping with $\sigma_v$ predicted by linear theory and allowing $\sigma_v$ to be a free, empirical parameter. The light red squares at $k_{max}=0.2 \,h$/Mpc are the result of fitting only over the range $0.1 \leq k< 0.2\,h$/Mpc.
• Figure 4: $P^s(k,\mu)$ for $z=0,0.5,1$ as in Fig. \ref{['fig:kaiser2D']} but calculated using the Scoccimarro formula with linear theory damping, and $P_{\theta\theta}$, $P_{\theta\delta}$ and $P_{\delta\delta}$ measured from simulations (green) or 1-loop SPT (dashed blue), compared to the redshift space power spectrum from N-body simulations (black, labelled contours).
• Figure 5: 1$\sigma$ constraints on the growth rates derived from modelling the redshift space power spectrum with SPT, LPT and Taruya$^{++}$ (with linear theory and empirical damping) models of the full redshift space power spectrum. We have considered the Taruya$^{++}$ model with both linear theory and empirical damping terms but the SPT model does not contain any damping. Values of $f$ obtained for SPT and LPT have not been shown at $z=0,0.5$ for $k_{max}=0.2$ because they were consistent with zero.