Merge pull request #319 from LSSTDESC/linlog_pk

Hybrid linear-logarithmic sampling on a for power spectrum

doc/0000-ccl_note/main.tex

@@ -422,9 +422,11 @@ \subsubsection{Spline parameters \& the INI file}
 \item {\tt LOGM$\_$SPLINE$\_$NM}: number of points in the mass function spline (logarithm). The default is {\tt LOGM$\_$SPLINE$\_$NA}$=440$.
 \item {\tt LOGM$\_$SPLINE$\_$MIN}: minimum value for the mass function spline (logarithm). The default is {\tt LOGM$\_$SPLINE$\_$MIN}$=6$.
 \item {\tt LOGM$\_$SPLINE$\_$MAX}: maximum value for the mass function spline (logarithm). The default is {\tt LOGM$\_$SPLINE$\_$MAX}$=17$.
-\item {\tt A$\_$SPLINE$\_$NA$\_$PK}: number of bins for the scale factor in the case of two-dimensional splines (where the second variable is the wavenumber). The default is {\tt A$\_$SPLINE$\_$NA$\_$PK}$=50$. For the emulator, this is overwritten by adopting {\tt K$\_$MAX$\_$SPLINE}$=5/$Mpc for the nonlinear matter power spectrum.  
-\item {\tt A$\_$SPLINE$\_$MIN$\_$PK}: minimum value for the scale factor spline. The default is {\tt A$\_$SPLINE$\_$MIN$\_$PK}$=0.1$. This only applies to the 2D power spectrum splines. For the cosmic emulator and in the case of the nonlinear matter power spectrum, this is overwritten by adopting {\tt A$\_$SPLINE$\_$MIN\_PK}$=1/3$.
-\item {\tt K$\_$MAX$\_$SPLINE}: The maximum value of wavenumber considered in the spline. This is explained in more detail in the coming subsections. The default is {\tt K$\_$MAX$\_$SPLINE}$=50/$Mpc.
+\item {\tt A$\_$SPLINE$\_$NLOG$\_$PK}: number of bins for the logarithmic part of the scale factor in the case of two-dimensional splines (where the second variable is the wavenumber). The default is {\tt A$\_$SPLINE$\_$NLOG$\_$PK}$=11$.
+\item {\tt A$\_$SPLINE$\_$NA$\_$PK}: number of bins for the linear part of the scale factor in the case of two-dimensional splines (where the second variable is the wavenumber). The default is {\tt A$\_$SPLINE$\_$NA$\_$PK}$=40$. In the case of the emulator's nonlinear power spectrum, the maximum wavenumber is {\tt K$\_$MAX$\_$SPLINE}$=5/$Mpc, the interpolation is only linear and we use {\tt A$\_$SPLINE$\_$NA$\_$PK}$=40$ as default. 
+\item {\tt A$\_$SPLINE$\_$MINLOG$\_$PK}: minimum value for the scale factor spline. The default is {\tt A$\_$SPLINE$\_$MINLOG$\_$PK}$=0.01$. This only applies for the 2D power spectrum splines. For the cosmic emulator and in the case of the nonlinear matter power spectrum, we use a linear spline only, adopting {\tt A$\_$SPLINE$\_$MIN\_PK}$=1/3$.
+\item {\tt A$\_$SPLINE$\_$MIN$\_$PK}: maximum value for the logarithmic part and minimum value for the linear part of the scale factor spline. The default is {\tt A$\_$SPLINE$\_$MIN$\_$PK}$=0.1$. This only applies for the 2D power spectrum splines.
+\item {\tt K$\_$MAX$\_$SPLINE}: The maximum value of wavenumber considered in the spline. This is explained in more detail in the coming subsections. The default is {\tt K$\_$MAX$\_$SPLINE}$=50/$Mpc. In the case of the emulator's nonlinear power spectrum, the maximum wavenumber is {\tt K$\_$MAX$\_$SPLINE}$=5/$Mpc.
 \item {\tt K$\_$MAX}: The maximum value of wavenumber when integrating over $k$. The default is {\tt K$\_$MAX}$=1000$/Mpc.
 \item {\tt K$\_$MIN$\_$DEFAULT}:  The minimum value of wavenumber when integrating over $k$. The default is {\tt K$\_$MIN$\_$DEFAULT}$=5 \times 10^{-5}$/Mpc.
 \item {\tt N$\_$K}: Number of bins per decade for the wavenumber. The default is {\tt N$\_$K}$=167$.
@@ -447,7 +449,7 @@ \subsubsection{Extrapolation for the nonlinear power spectrum}
   \label{eq:NLPSTaylor}
 \end{equation}
 
-The accuracy of this approximation is shown in Figure \ref{fig:NLextrapol} for redshifts $z=0$ and $z=3$. We compare the nonlinear matter power spectrum at $z=0$ computed with the previously described approximation, to the matter power spectrum obtained by directly evaluating {\tt CLASS} at the desired $k$ value. In figure \ref{fig:NLextrapol}, we vary $\Delta \ln k$ and {\tt K$\_$MAX$\_$SPLINE} at a time to determine the impact of the choice of parameters. We find that for typical values of $\Delta \ln k=10^{-2}$ and {\tt K$\_$MAX$\_$SPLINE}$=50$/Mpc, $\ln P$ has converged to an accuracy that surpasses the expected impact of baryonic effects on the matter power spectrum at $k>10/$Mpc.  (For an estimate of the impact of baryons on the total matter power spectrum, see \citealt{Schneider15}.) The lower {\tt K$\_$MAX$\_$SPLINE} is, the faster \ccl will run. The optimum choice of {\tt K$\_$MAX$\_$SPLINE} is left to the user for their particular application.
+The accuracy of this approximation is shown in Figure \ref{fig:NLextrapol} for redshifts $z=0$, $z=3$ and $z=20$. We compare the nonlinear matter power spectrum at these redshifts computed with the previously described approximation, to the matter power spectrum obtained by setting the power spectrum splines to high values. We find that for typical values of $\Delta \ln k=10^{-2}$ and {\tt K$\_$MAX$\_$SPLINE}$=50$/Mpc, $\ln P$ has converged to an accuracy that surpasses the expected impact of baryonic effects on the matter power spectrum at $k>10/$Mpc.  (For an estimate of the impact of baryons on the total matter power spectrum, see \citealt{Schneider15}.) The lower {\tt K$\_$MAX$\_$SPLINE} is, the faster \ccl will run. The optimum choice of {\tt K$\_$MAX$\_$SPLINE} is left to the user for their particular application.
 
 We also extrapolate the power spectrum at small wavenumbers within the static routine {\tt ccl\_power\_extrapol\_lowk}. In this case, the power spectrum below {\tt K$\_$MIN} is obtained by a power-law extrapolation with index $n_s$:
 \begin{equation}
@@ -459,29 +461,20 @@ \subsubsection{Extrapolation for the nonlinear power spectrum}
 %------------------------
 \begin{figure*}
 \centering
-\includegraphics[width=0.9\textwidth]{PS_converge_nonlin.eps}
-\caption{The relative error compared to direct CLASS outputs, $P_{fid}$, produced by splining the nonlinear matter power spectrum up to {\tt K$\_$MAX$\_$SPLINE} and extrapolating beyond this value with a second order Taylor expansion the natural logarithm of the matter power spectrum. The left panel shows the results at $z=0$. The right panel shows the results at $z=3$. The standard \ccl parameters adopted are those corresponding to the black dashed curve. The solid curve shows the accuracy when using {\tt K$\_$MAX$\_$SPLINE}$=500$/Mpc. The magenta curve shows the impact of using the higher resolution $\Delta\ln k=10^{-4}$ in Eq. \ref{eq:NLPSTaylor}. For comparison, the impact of baryonic physics on the matter power spectrum is $\sim 10\%$ at $k=1$/Mpc \citep{Schneider15}.}
+\includegraphics[width=1.0\textwidth]{plot_power.eps}
+\caption{The relative error compared to power spectra produced with high values of the power spectrum splines, $P_{fid}$, produced by splining the matter power spectrum up to {\tt K$\_$MAX$\_$SPLINE} and extrapolating beyond this value with a second order Taylor expansion the natural logarithm of the matter power spectrum. The left panel shows the relative errors for the linear matter power spectrum at $z=0$, $z=3$ and $z=20$. The right panel shows the results for the non-linear matter power spectrum at the same redshifts. The standard \ccl parameters adopted are those corresponding to the black dashed curve. For comparison, the impact of baryonic physics on the matter power spectrum is $\sim 10\%$ at $k=1$/Mpc \citep{Schneider15}.}
 \label{fig:NLextrapol}
 \end{figure*}
 %------------------------
 
 \subsubsection{Extrapolation for the linear power spectrum}
 \label{sec:Lextrapol}
 
-With the implementation described in the previous section, the power spectrum splines are initialized up to {\tt K$\_$MAX$\_$SPLINE}. This is also true for the linear matter power spectrum, which is used within \ccl in particular to obtain $\sigma_8$ (see Eq. \ref{eq:sigR}). We have tested here how the procedure described in the previous section affects the convergence of the linear matter power spectrum. We compare the fiducial \ccl output to the case where we set {\tt K$\_$MAX$\_$SPLINE}$=5\times 10^3/$Mpc. The result is shown in Figure \ref{fig:Lextrapol}. Although there is a significant difference ($\gtrsim 10\%$) between the linear power spectra at large $k$, we have confirmed that the difference in $\sigma_8$ is negligible. Nevertheless, for other applications that use the linear power spectrum, the user might need to increase the value of {\tt K$\_$MAX$\_$SPLINE}.
-
-%------------------------
-\begin{figure*}
-\centering
-\includegraphics[width=0.9\textwidth]{PS_converge_lin.eps}
-\caption{Same as Fig. \ref{fig:NLextrapol} but for the linear matter power spectrum at $z=0$ (left) and $z=3$ (right).}
-\label{fig:Lextrapol}
-\end{figure*}
-%------------------------
+With the implementation described in the previous section, the power spectrum splines are initialized up to {\tt K$\_$MAX$\_$SPLINE}. This is also true for the linear matter power spectrum, which is used within \ccl in particular to obtain $\sigma_8$ (see Eq. \ref{eq:sigR}). We have tested here how the procedure described in the previous section affects the convergence of the linear matter power spectrum. The result is shown in Figure \ref{fig:NLextrapol}. For some applications that use the linear power spectrum, the user might need to increase the value of {\tt K$\_$MAX$\_$SPLINE}.
 
 As in the previous section, the power spectrum at small wavenumber is extrapolated using a power-law. This extrapolation is performed below a fiducial value of {\tt K\_MIN\_DEFAULT}$=5\times 10^{-5}$.
 
-We have found that changing {\tt N$\_$A} to $200$, or changing the sampling of the wavenumber to $5000$ points, does not change the results presented in Figures \ref{fig:NLextrapol} and \ref{fig:Lextrapol} in this section.
+We have found that changing {\tt N$\_$A} to $200$, or changing the sampling of the wavenumber to $5000$ points, does not change the results presented in Figures \ref{fig:NLextrapol} in this section.
 
 \subsubsection{Wishlist for the future}
 \label{Pk_wishlist}