Towards an accurate model of the redshift space clustering of halos in the quasilinear regime

TL;DR

This paper develops a non-perturbative, scale-dependent Gaussian streaming model to accurately predict redshift-space clustering of halos in the quasilinear regime, identifying non-linear real-to-redshift space mappings and velocity-field non-linearities as key corrections. By combining a scale-dependent GSM with perturbation theory for real-space halo clustering and velocity statistics, the authors produce an analytic model that matches N-body halo redshift-space multipoles $\xi_{0,2,4}$ at the percent level for separations $s \gtrsim 25-40\,h^{-1}\mathrm{Mpc}$, depending on bias. They demonstrate that the monopole is robust down to $\sim10\,h^{-1}\mathrm{Mpc}$ when input statistics are accurate, while the quadrupole requires careful handling of $v_{12}(r)$ and higher-order terms, with bias driving the size of non-linear mapping corrections (scaling roughly as $b^3$). The work also confirms that the large-scale real-space halo bias determines the pairwise infall velocity amplitude to within a percent, validating a key assumption for extracting $f\sigma_8$ from redshift-space data. Overall, this approach provides a practical path to percent-level RSD modeling for current and upcoming surveys, while outlining areas (satellites, non-Gaussian small-scale velocities) for further refinement.

Abstract

Observations of redshift-space distortions in spectroscopic galaxy surveys offer an attractive method for measuring the build-up of cosmological structure, which depends both on the expansion rate of the Universe and our theory of gravity. Galaxies occupy dark matter halos, whose redshift space clustering has a complex dependence on bias that cannot be inferred from the behavior of matter. We identify two distinct corrections on quasilinear scales (~ 30-80 Mpc/h): the non-linear mapping between real and redshift space positions, and the non-linear suppression of power in the velocity divergence field. We model the first non-perturbatively using the scale-dependent Gaussian streaming model, which we show is accurate at the <0.5 (2) per cent level in transforming real space clustering and velocity statistics into redshift space on scales s>10 (s>25) Mpc/h for the monopole (quadrupole) halo correlation functions. We use perturbation theory to predict the real space pairwise halo velocity statistics. Our fully analytic model is accurate at the 2 per cent level only on scales s > 40 Mpc/h. Recent models that neglect the corrections from the bispectrum and higher order terms from the non-linear real-to-redshift space mapping will not have the accuracy required for current and future observational analyses. Finally, we note that our simulation results confirm the essential but non-trivial assumption that on large scales, the bias inferred from real space clustering of halos is the same one that determines their pairwise infall velocity amplitude at the per cent level.

Figures (12)

• Figure 1: Contours of constant $\xi$ as a function of separation perpendicular (R) and parallel (Z) to the line-of-sight (LOS) from the mock galaxy catalogues in white/etal:2011. The magnitude of the squashing along the Z axis depends on the amplitude of the peculiar velocity field, which is related to the over-density field on large scales (Eq. \ref{['eq:thetadeltalin']}). The stretching of the contours near R=0 results from small-scale velocity dispersions and is often referred to as the 'fingers-of-god.'
• Figure 2: Comparison of $N$-body simulation redshift space clustering of halo samples in Table \ref{['table:halos']} with the biased Lagrangian perturbation theory of Mat08b. The high (circles) and low (triangles) halo mass bins are shifted by 10 per cent for clarity compared with the HOD subsample (diamonds) in the upper left panel. The predictions of Mat08b are shown as solid black curves; higher bias values have larger deviations from linear theory expectations (dashed curves) in LPT. While Mat08b provides an excellent fit to the simulation results for $\xi_0$ and $\xi_2$ on BAO scales (bottom right panel), the quasilinear scale description is not accurate.
• Figure 3: The radial pairwise halo velocity probability distribution function for the HOD halo subsample in Table \ref{['table:halos']} for pair separations $30\,h^{-1}$Mpc $< r < 31.5\,h^{-1}$Mpc (solid curves) compared with a Gaussian distribution (dashed curves) with the same mean ($-1.6\,h^{-1}$Mpc) and variance $18\,(h^{-1}$Mpc$)^2$. The normalisation is arbitrary. While the halo PDF has clear skewness and kurtosis, the PDF for dark matter particles in our simulation has 30 per cent larger variance and exponential tails. The mean infall between dark matter particles ($-0.9\,h^{-1}$Mpc) is smaller than the more highly biased halo samples.
• Figure 4: The solid curves show the linear theory Kaiser formula predictions for the Legendre moments of the redshift space correlation function, $\xi_{0,2,4}(s)$ given in Eq. \ref{['eq:ximoments']}, where $s$ is the redshift space pair separation for the three different bias values given in Table \ref{['table:halos']}, as well as $b=1$ and $b=0.5$ for comparison. The dashed lines show the predictions of the scale-dependent Gaussian streaming model, Eq. \ref{['streamingeqn']}, which we evaluated using the linear theory expectations for $\xi^{r}_g(r)$, $v_{12}(r)$, and $\sigma_{12}^2(r, \mu)$. We see that accounting for the full real-to-redshift space mapping for a Gaussian pairwise velocity probability distribution that agrees with linear theory significantly modifies $\xi_2$ and $\xi_4$ from the Kaiser formula expectation on quasilinear scales, with larger corrections at higher bias.
• Figure 5: Fractional error on $f\sigma_8$ and $b\sigma_8$ as a function of the minimum configuration space separation $s_{min}$ used in the analysis, assuming a covariance matrix given by the usual Gaussian cosmic variance term and Poisson sampling variance term, with $b=2$, $V_{survey} = 5 \,h^{-3}$Gpc$^{3}$, and $\bar{n} = 3.0 \times 10^{-4} \,h^{3}$Mpc$^{-3}$. We use the model in Eq. \ref{['eq:pkgauss']}, and consider two cases. In the first, we marginalise over the value of $\sigma$, which controls the small scale damping. In the second case, we assume $\sigma$ is perfectly known; in both the fiducial value of $\sigma$ is $3.5\,h^{-1}$Mpc. Marginalisation over $\sigma$ increases the error on $b\sigma_8$ and $f\sigma_8$ (upper two curves in both panels). The dashed curves show constraints when only $\xi_{0,2}$ are used; the solid curves include $\xi_4$ as well. In the case where $\sigma$ is known (lower curves), $\ell_{max} = 2$ and 4 are indistinguishable. Finally, we compare the configuration space results to an analysis in Fourier space WhiSonPer09, where a sharp cut-off in wavenumber is imposed, shown as dotted curves as a function of $1.15\pi/k_{max}$ and also assuming $\sigma$ is known.