short section on halo mass function derivatives and interpolation
@@ -708,6 +708,12 @@ For achieving $10^{-4}$ precision in $\sigma(M)$ and the normalisation of the po
Also note that for $\sigma(M)$, it is important to set the desired precision level correctly for the numerical integrator. The integral usually yields $\sigma^2(M)$, and not $\sigma(M)$. Hence, one has to set the desired precision taking the exponent into account.
+Derivatives are calculated utilizing a spline that is built off of the previously described $\sigma(M)$ spline. For each value of $\log(M)$ in our spline evaluation, we calculate the value of $\sigma(M)$ half a step in either direction. We use the difference compared to the mass spacing to calculate an approximate derivative, which is then used in the spline interpolation. This has been tested to meet our necessary precision for the halo mass function.
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+In order to accomodate a wide range of values of the overdensity parameter $\Delta$, we have generated a spline interpolation between best fit values as defined by \citet{Tinker2008} and \citet{Tinker2010}. This covers a dynamic range from $\Delta=200$ to $3200$. Within this range, we interpolate in the space of the fit parameter and $\log\Delta$ using the Akima interpolation built from piecewise third order polynomials. We have chosen this rather than the fitting formulas utilized in \citet{Tinker2010} in order to assure high precision match to the Tinker halo mass function when choosing a value of $Delta$ directly from the paper.
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\subsection{Massive neutrinos}
\vol{Danielle Leonard}
