= Limber approximation
{c}
{title2=$C_\ell^{uv}\simeq\int d\chi\,q_uq_vP_{UV}((\ell+1/2)/\chi,z)/\chi^2$}
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.
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