Limber approximation ($C_{\ell }^{uv}\simeq \int d\chi q_u{q}_v{P}_{UV}((\ell +1/2)/\chi ,z)/{\chi }^2$)

For fields projected along the line of sight, the Limber approximation replaces the oscillatory radial Spherical Bessel function coupling by a local radial contribution with $k\chi \simeq \ell +1/2$. The displayed cross-power spectrum formula applies to $u(\hat{\mathbf{n}})=\int d\chi q_u(\chi )U(\chi \hat{\mathbf{n}},z)$ and similarly for $v$, in a spatially flat background. It requires radial kernels and the three-dimensional cross-power spectrum to vary slowly across the radial coupling width. It suppresses correlations of disjoint radial kernels but can fail at low multipoles or for very narrow selections; exact unequal-time correlations can remain nonzero there.