Redshift Distortions of Clustering: A Lagrangian Approach

TL;DR

This paper develops a Lagrangian perturbation approach to analyze redshift distortions in galaxy clustering, providing explicit formulas for skewness and bispectrum, and validating them against simulations for improved large-scale structure analysis.

Contribution

It introduces a second-order Lagrangian perturbation framework to accurately model redshift space skewness and bispectrum, extending previous first-order solutions and including effects of smoothing.

Findings

Analytical expressions match numerical simulations well.

Second-order theory improves accuracy over first-order models.

Results can aid in interpreting large-scale redshift surveys.

Abstract

We study the effects of peculiar velocities on statistical measures of galaxy clustering. These effects occur when distances to the galaxies are estimated from their redshifts. It is assumed that the clustering pattern results from the gravitational instability of initially Gaussian, small-amplitude perturbations of a Friedman-Lemaitre cosmological model. Explicit expressions are given for an arbitrary density parameter \Omega of the model, both when the cosmological constant, \Lambda, is zero, and when the model is spatially flat, \Omega + \Lambda/3H^2 =1. We show how to compute the skewness, or third moment of the density field, in redshift space, and its Fourier space counterpart, the bispectrum. We rely on a perturbative expansion of particle trajectories in Lagrangian coordinates. This formalism extends to higher orders the Zel'dovich first order solution (1970). We show that a physically consistent and quantitatively accurate analysis of the growth skewness in redshift space can be obtained from second-order Lagrangian theory. We also study the effects of spatial smoothing of the evolved density field. We give analytic expressions for the gravitationally induced skewness as a function of the power spectrum and of \Omega, for a spherical top-hat and a Gaussian smoothing filter. We compare our analytical predictions with measurements performed in numerical simulations, and find good agreement. These results should then prove useful in analyzing large scale redshift surveys. In particular, our results, in conjunction with the recent suggestion of Fry (1994), may help to break the coupling between the bias and the cosmological