On The Validity of the Streaming Model for the Redshift-Space Correlation Function in the Linear Regime

The relation between the galaxy correlation function in real and redshift-space is derived in the linear regime by an appropriate averaging of the joint probability distribution of density and velocity. The derivation recovers the familiar linear theory result on large scales but has the advantage of clearly revealing the dependence of the redshift distortions on the underlying peculiar velocity field; streaming motions give rise to distortions of ${\cal O}(\Omega^{0.6}/b)$ while variations in the anisotropic velocity dispersion yield terms of order ${\cal O}(\Omega^{1.2}/b^2)$. This probabilistic derivation of the redshift-space correlation function is similar in spirit to the derivation of the commonly used ``streaming'' model, in which the distortions are given by a convolution of the real-space correlation function with a velocity distribution function. The streaming model is often used to model the redshift-space correlation function on small, highly non-linear, scales. There have been claims in the literature, however, that the streaming model is not valid in the linear regime. Our analysis confirms this claim, but we show that the streaming model can be made consistent with linear theory {\it provided} that the model for the streaming has the functional form predicted by linear theory and that velocity distribution is chosen to be a Gaussian with the correct linear theory dispersion.