Redshift distortions of clustering : a Lagrangian approach

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–Lemaˆıtre cosmological model. Explicit expressions are given for an arbitrary density parameter of the model, both when the cosmological constant, , is zero, and when the model is spatially flat, + =3H 2 = 1. Kaiser (1987) had analyzed the redshift distortion of the two–point correlation function. This function determines the variance of the density field distribution function and can be computed using linear perturbation theory. We show here how to compute higher order moments in redshift space, paying special attention to the skewness, or third moment of the density field, and its Fourier space counterpart, the bispectrum. This calls for a (weakly) non–linear analysis. We rely on a perturbative expansion of particle trajectories in Lagrangian coordinates, using the formalism introduced by Moutarde et al. (1991) and further developed by Bouchet et al. (1992, 1994). This formalism extends to higher orders the Zel'dovich first order (i.e. linear) solution (1970). The lowest non-vanishing contribution to the skewness comes from the first and second-order terms in perturbation theory. Therefore, using Zel'dovich approximation would not be self-consistent and would yield inaccurate results. 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. With practical applications to redshift surveys in mind, we also study the effects of spatial smoothing of the evolved density field. The necessary formalism was developed by Juszkiewicz and Bouchet (1991) and Juszkiewicz et al. (1993a). Here we give the first complete account of these calculations; we also extend the formalism by explicitly taking redshift distortions into account. We give analytic expressions for the gravitation-Send offprint requests to: E. Hivon ally induced skewness as a function of the power spectrum and of , 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 solve a well known problem which always arises in conventional dynamical deter-minations of the mean density of the universe. …