Fast, Accurate and Perturbative Forward Modeling of Galaxy Clustering Part II: Redshift Space

TL;DR

This work extends the LEFTfield perturbative forward model to redshift space by introducing a one-step displacement from Lagrangian space to redshift space, incorporating an expanded Lagrangian bias and velocity-bias framework. It provides a detailed assessment of perturbative and numerical errors for momentum, velocity, and redshift-space density, and demonstrates growth-rate inference at fixed initial conditions with percent-level accuracy and modest systematic shifts. The model achieves ~1.5× computational speed-up over the rest-frame version, enabling efficient field-level analyses and simulation-based inference in redshift space. The results validate the viability of field-level, redshift-space EFT analyses for upcoming surveys, while outlining avenues for improved noise modeling and light-cone implementations.

Abstract

Forward modeling the galaxy density within the Effective Field Theory of Large Scale Structure (EFT of LSS) enables field-level analyses that are robust to theoretical uncertainties. At the same time, they can maximize the constraining power from galaxy clustering on the scales amenable to perturbation theory. In order to apply the method to galaxy surveys, the forward model must account for the full observational complexity of the data. In this context, a major challenge is the inclusion of redshift space distortions (RSDs) from the peculiar motion of galaxies. Here, we present improvements in the efficiency and accuracy of the RSD modeling in the perturbative LEFTfield forward model. We perform a detailed quantification of the perturbative and numerical error for the prediction of momentum, velocity and the redshift-space matter density. Further, we test the recovery of cosmological parameters at the field level, namely the growth rate $f$, from simulated halos in redshift space. For a rigorous test and to scan through a wide range of analysis choices, we fix the linear (initial) density field to the known ground truth but marginalize over all unknown bias coefficients and noise amplitudes. With a third-order model for gravity and bias, our results yield $<1\,\%$ statistical and $<1.5\,\%$ systematic error. The computational cost of the redshift-space forward model is only $\sim 1.5$ times of the rest frame equivalent, enabling future field-level inference that simultaneously targets cosmological parameters and the initial matter distribution.

Figures (32)

• Figure 1: Perturbative accuracy of the forward model for the momentum along the LOS-direction, $\pi_\parallel$, at $z=0.5$. We compare results from the forward model (subscript "f") to data from a N-body simulation which has an identical Fourier-space top-hat filter at $\Lambda=0.20\,h/\mathrm{Mpc}$ applied to its initial conditions (subscript "d"). Clockwise, starting from the top left panel, we show the power spectrum difference, the residual power spectrum, the cross-correlation between forward model and simulation and the cross-correlation between residuals and the simulation. For LPT orders $n_\mathrm{LPT} \geq 3$, we also explore the impact of the transverse displacement component and find it to be negligible. The 4LPT model predicts the momentum power spectrum to better than 4% accuracy. See figure \ref{['fig:velocity-accuracy__accuracy-velocity-momentum-density_L010_z050']} for results at $\Lambda=0.10\,h/\mathrm{Mpc}$ and figure \ref{['fig:velocity-accuracy__accuracy-velocity-momentum-density_z100']} for $z=1.0$.
• Figure 2: Slices through the LOS-velocity $v_\parallel$ at $z=0.0$ for a cut-off $\Lambda=0.20\,h/\mathrm{Mpc}$, where the LOS direction is perpendicular to the image plane. In the top row, we show velocity grids computed by the Delaunay tessellation for a grid size of $N_\mathrm{G,D}=128$ from the reference simulation (left) and from the 3LPT forward model at increasing displacement resolution $N_\mathrm{G,Eul}$. We compare these results to velocity grids obtained by dividing out the density from the momentum in the bottom panels. Figure \ref{['fig:velocity-accuracy__presentation__grid-comparison-L010']} extends the results to a lower cut-off, $\Lambda=0.10\,h/\mathrm{Mpc}$.
• Figure 3: Perturbative accuracy of the forward model for the velocity along the LOS-direction, $u_{\parallel}$, at $z=0.5$. We compare results from the forward model (subscript "f") to data from the reference simulation (subscript "d"); both use identical initial conditions which are filtered at $\Lambda=0.20\,h/\mathrm{Mpc}$. The velocity is computed on a grid of size $N_\mathrm{G,D}=256$ from simulations with $N_\mathrm{part.}=1536$ and the forward model with $N_\mathrm{part.}=1024$ using the Delaunay tessellation algorithm. Gray lines indicate the expected difference due to the the different number of particles in forward model and simulations (see appendix \ref{['sec:delaunay']}), which are subdominant. Clockwise, starting from the top left panel, we show the power spectrum difference, the residual power spectrum, the cross-correlation between forward model and simulation and the cross-correlation between residuals and the simulation. For LPT orders $n_\mathrm{LPT} \geq 3$, we also explore the impact of the transverse displacement component and find it to be negligible. The size of the residuals relative to the reference simulation are comparable for velocities and momentum. See figure \ref{['fig:velocity-accuracy__accuracy-velocity-momentum-density_L010_z050']} for results at $\Lambda=0.10\,h/\mathrm{Mpc}$ and figure \ref{['fig:velocity-accuracy__accuracy-velocity-momentum-density_z100']} for $z=1.0$.
• Figure 4: Measurement of the deterministic (left) and stochastic (right) counter terms from the comparison of the forward model (cut-off $\Lambda=0.20\,h/\mathrm{Mpc}$) with N-body simulations (no initial cut-off). The model predictions are evaluated at 4th order in LPT and include the transverse contribution to the displacement. The bump in the momentum and velocity transfer function is connected to the perturbative accuracy of the forward model and only present for low redshifts and high cut-offs; it vanishes for $\Lambda=0.10\,h/\mathrm{Mpc}$. Apart from that, we recover the expected leading-order $k$-scaling for all terms. Note that, in principle, a white noise contribution would be allowed for the velocities. However, it appears to be highly suppressed at least in dark-matter-only simulations.
• Figure 5: Forward model of the galaxy density field $\tilde{\delta}_{\mathrm{g},\mathrm{det}}$ in redshift space. The model requires only a single displacement from the Lagrangian frame to redshift space by adding up the LPT displacement vector and the tracer velocity $u_{\mathrm{g},\mathrm{det}\parallel}$. The latter includes higher-order bias operators $\{\mathcal{U}\}$. The density bias operators $\{\mathcal{O}_{\mathrm{L}}\}$ are constructed in the Lagrangian frame and then displaced to redshift space. In the middle column, we list the parameters which control the numerical accuracy of the forward model. Explanations and best-practice recommendations are given in the right column (see also ref. Stadler:2024).