Extended Limber approximation

We develop a systematic derivation for the Limber approximation to the angular cross-power spectrum of two random fields, as a series expansion in $(\ensuremath{\ell}+1/2{)}^{\ensuremath{-}1}$. This extended Limber approximation can be used to test the accuracy of the Limber approximation and to improve the rate of convergence at large $\ensuremath{\ell}$'s. We show that the error in ordinary Limber approximation is $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}2})$. We also provide a simple expression for the 2nd order correction to the Limber formula, which improves the accuracy to $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}4})$. This correction can be especially useful for narrow redshift bins, or samples with small redshift overlap, for which the 0th order Limber formula has a large error. We also point out that using $\ensuremath{\ell}$ instead of $\ensuremath{\ell}+1/2$, as is often done in the literature, spoils the accuracy of the approximation to $\mathcal{O}({\ensuremath{\ell}}^{\ensuremath{-}1})$.