Update main.tex
Change reference  2017arXiv170103592C into published version 2017A&A...602A..72C (JEC)
@@ -680,7 +680,7 @@ The Limber approximation works best for wide radial kernels and high $\ell$.
\ccl incorporates natively routines to compute the $C^{ab}_\ell$ angular power spectra as described above without the Limber approximation. The algorithm performs first the integrals over $z$ for both tracers, and ends with the $k$ integral. This computation is much slower than using the Limber approximation, but results in precise angular power spectra at low $\ell$, and correct cross-correlations between tracers. Indeed as the Limber approximation consists in replacing oscillatory functions by Dirac distributions, the latter fails at reproducing the beat phenomenon $j_\ell(x)\times j_\ell(x')$ appearing when two oscillatory functions are multiplied. The integration of these routines has been tested against the \texttt{CLASS} code and recovers the same angular power spectra when precision parameters are set to high values.
-However, the computation of the $C^{ab}_\ell$ without the Limber approximation is too costly in terms of computing time using this method, if one wants to explore extensively a full cosmological parameter space. The aim of the \texttt{Angpow} software \citep{2017arXiv170103592C} is to compute the angular power spectra $C_{\ell}^{ab}$ without any Limber numerical approximation in a faster way but still accurately. \ccl has been linked to the \texttt{Angpow} code, which is briefly described here.
+However, the computation of the $C^{ab}_\ell$ without the Limber approximation is too costly in terms of computing time using this method, if one wants to explore extensively a full cosmological parameter space. The aim of the \texttt{Angpow} software \citep{2017A&A...602A..72C} is to compute the angular power spectra $C_{\ell}^{ab}$ without any Limber numerical approximation in a faster way but still accurately. \ccl has been linked to the \texttt{Angpow} code, which is briefly described here.
The angular power spectrum for two tracers $C_{\ell}^{ab}$ is computed in \texttt{Angpow} according to the following expression
\begin{equation}
@@ -699,7 +699,7 @@ with $j_\ell(x)$ and $j_\ell^{\prime\prime}(x)$ the spherical Bessel function of
%\end{itemize}
In \texttt{Angpow}, the inner integral in $k$ is computed first.
-To conduct the computation of such an integral of highly oscillating functions, the 3C-algorithm described in details in \citep{2017arXiv170103592C} is used. In brief, it relies on the projection of the oscillating $f_\ell(z_,k)$ onto a Chebyshev series of order $2^N$, the product of the two Chebyshev series is performed with a $2^{2N}$ Chebyshev series; then the integral is computed using Clenshaw-Curtis quadrature. Finally, the integrals over $z$ are performed via again an optimised Clenshaw-Curtis quadrature. All the Chebyshev expansions and the Clenshaw-Curtis quadrature are performed via the \textit{Discrete Cosine Transform} of type I from the DCT-I fast transform of the FFTW library \citep{FFTW}.
+To conduct the computation of such an integral of highly oscillating functions, the 3C-algorithm described in details in \citep{2017A&A...602A..72C} is used. In brief, it relies on the projection of the oscillating $f_\ell(z_,k)$ onto a Chebyshev series of order $2^N$, the product of the two Chebyshev series is performed with a $2^{2N}$ Chebyshev series; then the integral is computed using Clenshaw-Curtis quadrature. Finally, the integrals over $z$ are performed via again an optimised Clenshaw-Curtis quadrature. All the Chebyshev expansions and the Clenshaw-Curtis quadrature are performed via the \textit{Discrete Cosine Transform} of type I from the DCT-I fast transform of the FFTW library \citep{FFTW}.
As in the general case the Limber approximation is valid at high $\ell$ values, the \ccl user can define an $\ell$ threshold to switch from the non-Limber slow computation to the faster Limber approximation.
