How accurate is Limber ’ s equation ?

Aims. The so-called Limber equation is widely used in the literature to relate the projected angular clustering of galaxies to the spatial clustering of galaxies in an approximate way. This note gives estimates of where the regime of applicability of Limber's equation stops. Methods. This paper revisits Limber's equation, summarises its underlying assumptions, and compares its predictions to the accurate integral of which the Limber equation is an approximation for some realistic situations. Thereby, the accuracy of the approximation is quantified. Different cases of spatial correlation functions are considered. Results. Limber's equation is accurate for small galaxy separations but breaks down beyond a certain separation that depends mainly on the ratio σ/rm and to some degree on the power-law index of spatial clustering; σ is the 1σ-width of the galaxy distribution in comoving distance, and rm the mean comoving distance. Limber's equation introduces a 10% systematic error that may be important at a scale of about a few degrees or in some cases, narrow galaxy distributions for example, even at arcmin scales. As rule-of-thumb, a 10% relative error is reached at ∼ 260 σ/rm arcmin for γ ∼ 1.6, if the spatial clustering is a power-law ξ ∝ r −γ. More realistic ξ are discussed in the paper. Conclusions. Limber's equation becomes increasingly inaccurate for larger angular separations. Ignoring this effect and blindly applying Limber's equation can possibly bias results for the inferred spatial correlation. It is suggested to use in cases of doubt, or maybe even in general, the exact equation that can easily be integrated numerically in the form given in the paper. This would actually eliminate all problems discussed in this paper.