Non-linear redshift distortions: The two-point correlation function

Abstract

We consider a situation where the density and peculiar velocities in real space are linear, and we calculate ξs the two-point correlation function in redshift space, incorporating all non-linear effects which arise as a consequence of the map from real to redshift space. Our result is non-perturbative and it includes the effects of possible multi-streaming in redshift space. We find that the deviations from the predictions of the linear redshift distortion analysis increase for the higher spherical harmonics of ξs. While the deviations are insignificant for the monopole ξ0, the hexadecapole ξ4 exhibits large deviations from the linear predictions. For a COBE normalized Γ = 0.25, h = 0.5 cold dark matter (CDM) power spectrum, our results for ξ4 deviate from the linear predictions by a factor of two on the scale of ∼10 h-1 Mpc. The deviations from the linear predictions depend separately on f(Ω) and b. This holds the possibility of removing the degeneracy that exists between these two parameters in the linear analysis of redshift surveys which yields only β= f(Ω)/b. We also show that the commonly used phenomenological model, where the non-linear redshift two-point correlation function is calculated by convolving the linear redshift correlation function with an isotropic pair velocity distribution function, is a limiting case of our result.