added small footnote regarding mass function validation and choice of single cosmology
@@ -945,7 +945,7 @@ 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. 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 hold to $<10^{-4}$ 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 cosmnology.}. 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 hold to $<10^{-4}$ tolerance level.
\begin{figure*}
\includegraphics[width=0.32\textwidth]{hmf_model1}
