[1003.4282] Modelling redshift space distortions in hierarchical cosmologies

Authors:  Elise Jennings, Carlton M. Baugh, Silvia Pascoli
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 $\Lambda$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 \citet{Kaiser:1987qv} is a poor fit to the measured distortions, even on surprisingly large scales $k \sim 0.03 h$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 10% 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.
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Fergus Simpson
Posts: 27
Joined: September 25 2004
Affiliation: University of Barcelona

[1003.4282] Modelling redshift space distortions in hierarc

Post by Fergus Simpson » March 25 2010

This interesting work looks at redshift space distortions on large scales, using results from N-body simulations. The central result is an improved prescription for modelling the influence of nonlinearities. It's been anticipated for a while that as more precise measures of the growth rate are sought after, we'd have to improve on the standard linear form

[tex]P(k,\mu) = (1+\beta \mu^2)^2 P(k)[/tex]

though most theorists have been reluctant to leave this behind as it's nice and easy to work with.

The prescription here is based on the fitting function presented in eq (15). Given the nonlinear matter power spectrum, this generates the velocity-velocity and velocity-density power spectra at [tex]z=0[/tex] , which may then be scaled to higher redshifts using (17) and (18). Finally, the model by Scoccimarro (10) may be used to generate the full [tex]P(k,\mu)[/tex].

Just a few questions:

- When applying (15), is "z=0" effectively meaning "[tex]\sigma_8 = 0.8[/tex], b=1"? In other words, are the alpha coefficients not dependent on [tex]\sigma_8[/tex] or the bias? Though if so, presumably a rescaling argument similar to (18) could be applied.

- Could we reach the point where these departures from linearity provide extra information, and improve constraints on the growth of structure?

- Looking at (15), don't some of the alpha coefficients require dimensions?

My only worry is Figure 4, which looks at how the estimated power spectrum is sensitive to the chosen gridding of the density/velocity fields. At first glance the lines seem to converge for increasing N, however the rate of increase in N is also diminishing! The fractional increments between the four lines are 100%, 37%, and 4%. So it looks like the line could still move a reasonable distance (~10%?) from the N=350 line, before we close in on the true N->\infty value.

Elise Jennings
Posts: 1
Joined: March 25 2010
Affiliation: Durham University

[1003.4282] Modelling redshift space distortions in hierarc

Post by Elise Jennings » March 26 2010

Hi Fergus,

Thank you for your interest in our paper. The answers to your questions are below.

1. Equation (15) is for [tex]b=1[/tex] and [tex]\sigma_8 =0.8[/tex]. So far we have only checked the density-velocity relation for dark matter and in all cosmologies the [tex]P(k)[/tex] have been normalised to the same [tex]\sigma_8[/tex] today. Although previous work, which we discuss in our paper, has shown this relation is independent of cosmological parameters so the dependence on [tex]\sigma_8[/tex] might not be very strong. The quintessence simulations have different [tex]\Omega_{\rm m}[/tex], [tex]\Omega_{\rm b}[/tex] and [tex]H_0[/tex] compared to [tex]\Lambda[/tex]CDM as in 0908.1394

2. Yes, some of these constants are dimensionful, this will be corrected when I replace the arxiv version with the published version.

3. We discuss the convergence seen in Fig. 4 in the text on page 9. Pueblas & Scoccimarro 2009 (0809.4606) compared tessellation and CIC assignment schemes in their work and using their results as a guide we can be sure our results have converged on scales up to [tex]k \sim 0.2 h/[/tex]Mpc. Including the velocity divergence power spectra on these scales in the model of the redshift space [tex]P(k,\mu)[/tex] is an improvement over the Kaiser 1987 linear perturbation theory.

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