added possible source of numerical difference for halo mass function
@@ -952,7 +952,7 @@ We define three new accuracy metrics:
\end{eqnarray}
Note that for $\sigma(M)$, it is important to set the desired precision level correctly for the numerical integrator. As the integral yields $\sigma^2(M)$, this becomes the relevant concern for numerical accuracy.
-For $\mathcal{A}_{hmf1}$ and $\mathcal{A}_{hmf3}$ we achieve accuracies of $3\times 10^{-5}$ and $5 \times 10^{-5}$, respectively. For $\mathcal{A}_{hmf2}$, the accuracy degrades to a value of $10^{-3}$. These accuracy levels are acceptable, as it is significantly better than the physical accuracy of current halo mass function models. This is demonstrated in Figure~\ref{fig:hmf}, where this calculation has been run for a single cosmology using the \citet{Tinker2010} halo mass function\footnote{A single cosmology is used for this analysis as the Tinker fitting parameters do not vary with cosmology.}. While there is a degradation in accuracy due to our spline treatment of the log inverse of $\sigma(M)$, we note that it does not significantly degrade our halo mass function determination. While improvement on this remains a task for the future, the halo mass function varies between fitting functions significantly more than this remaining error. As of this time, we do not have independent implementations for the halo bias function, though it should be noted that this calculation does not involve any additional functions beyond $\sigma(M)$ and should not exceed a $10^{-4}$ tolerance level.
+For $\mathcal{A}_{hmf1}$ and $\mathcal{A}_{hmf3}$ we achieve accuracies of $3\times 10^{-5}$ and $5 \times 10^{-5}$, respectively. For $\mathcal{A}_{hmf2}$, the accuracy degrades to a value of $10^{-3}$. These accuracy levels are acceptable, as it is significantly better than the physical accuracy of current halo mass function models. This is demonstrated in Figure~\ref{fig:hmf}, where this calculation has been run for a single cosmology using the \citet{Tinker2010} halo mass function\footnote{A single cosmology is used for this analysis as the Tinker fitting parameters do not vary with cosmology.}. While there is a degradation in accuracy due to our spline treatment of the log inverse of $\sigma(M)$, we note that it does not significantly degrade our halo mass function determination. While improvement on this remains a task for the future, the halo mass function varies between fitting functions significantly more than this remaining error. As of this time, we do not have independent implementations for the halo bias function, though it should be noted that this calculation does not involve any additional functions beyond $\sigma(M)$ and should not exceed a $10^{-4}$ tolerance level. {\bf Some deviation may exist between \ccl and other implementations due to our choice of spline interpolation between Tinker fitting parameters as a function of $\Delta$; we use an Akima interpolation between those provided in \citet{Tinker2010} for the halo mass function and this may lead to mild numerical change. This approach is motivated by the fact that the difference in parameters at $\Delta = 200$ between the fitting formula result and the tabulated best-fit parameters leading to a greater than $10^{-4}$ error from our calculated benchmarks in the halo mass function. As the tabulated version is in common use in the literature, having higher accuracy for the tabulated points was prioritized.}
\begin{figure*}
\includegraphics[width=0.32\textwidth]{hmf_model1}
