The Correlation Function in Redshift Space: General Formula with Wide-angle Effects and Cosmological Distortions

Abstract

A general formula for the correlation function in redshift space is derived in linear theory. The formula simultaneously includes wide-angle effects and cosmological distortions. The formula is applicable to any pair with arbitrary angle $θ$ between lines of sight, and arbitrary redshifts, $z_1$, $z_2$, which are not necessarily small. The effects of the spatial curvature both on geometry and on fluctuation spectrum are properly taken into account, and thus our formula holds in a Friedman-Lema\^ıtre universe with arbitrary cosmological parameters $Ω_0$ and $λ_0$. We illustrate the pattern of the resulting correlation function with several models, and also show that validity region of the conventional distant observer approximation is $θ\le 10^\circ$.