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 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/r_m 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 r_m 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. An alternative integral for such cases is given inside this paper.