diff for non-limber computations with CLASS benchmarks
@@ -1084,7 +1084,7 @@ differences at the levels observed are not surprising.
%-------------------------
%Angpow
-\ccl performs non-Limber computations of angular power spectra through the {\tt Angpow} library as detailed in Section \ref{sec:angpow}. The \texttt{Angpow} software was tested against \class and the brute-force integration \ccl computation, and can perform the same computations approximately an order of magnitude faster ($\mathcal{O}(1s)$). Its precision and speed parameters were optimised so that the relative numerical difference between the non-Limber computations is lower than two orders of magnitude, from $\ell=2$ to $\ell=1000$. We demonstrate this in Figure \ref{fig:angpow}, where we plot the angular clustering power spectrum for a sample of galaxies with $\langle z \rangle=1$ and a Gaussian redshift distribution that extends between $|z-\langle z \rangle|<5\sigma_z$, where $\sigma_z=0.02$, for a CCL1 cosmology. The non-Limber prediction deviates from the Limber case at low $l$ as expected. The right panel shows the fractional difference between the non-Limber curves, demonstrating the accuracy of the {\tt Angpow} prediction for our choice of precision and speed parameters. Also the brute-force non-Limber computation and {\tt Angpow} were tested to recover the Limber approximated curve at high $\ell$ for a wide Gaussian window ($\sigma_z=0.1$). The relative errors with respect to the Limber result at high $\ell$ are also lower than two orders of magnitudes compared to the expected cosmic variance.
+\ccl performs non-Limber computations of angular power spectra through the {\tt Angpow} library as detailed in Section \ref{sec:angpow}. The \texttt{Angpow} software was tested against \class and the brute-force integration \ccl computation, and can perform the same computations approximately an order of magnitude faster ($\mathcal{O}(1s)$). Its precision and speed parameters were optimised so that the relative numerical difference between the non-Limber computations is lower than two orders of magnitude, from $\ell=2$ to $\ell=1000$. We demonstrate this in Figure \ref{fig:angpow}, where we plot the angular clustering power spectrum for a sample of galaxies with $\langle z \rangle=1$ and a Gaussian redshift distribution that extends between $|z-\langle z \rangle|<5\sigma_z$, where $\sigma_z=0.02$, for a CCL1 cosmology. The non-Limber prediction deviates from the Limber case at low $l$ as expected. The right panel shows the fractional difference between the non-Limber curves, demonstrating the accuracy of the {\tt Angpow} prediction for our choice of precision and speed parameters. Also the brute-force non-Limber computation and {\tt Angpow} were tested to recover the Limber approximated curve at high $\ell$ for a wide Gaussian window ($\sigma_z=0.1$). The relative errors with respect to the Limber result at high $\ell$ are also lower than two orders of magnitudes compared to the expected cosmic variance. {\bf The differences between the \ccl~results and benchmarks from \class can be due to the integration methods and in particular the choice of the integral cut-off for small scales. The CCL Angpow implementation sets automatically the $k$ integral cut-off using the user-defined maximum multipole $\ell_{\rm max}$ as $k_{\rm max}=\pi \ell_{\rm max] / \chi{z_{\rm min}}$ where \chi{z_{\rm min}} is the minimum comoving distance within the redshift shells, while the \class user has to set it appropriately.}
%-------------------------------
\begin{figure*}[htbp]
