few more changes to 2.6
@@ -412,7 +412,7 @@ In the future \ccl will be expanded to incorporate the calculation of the higher
%-------------------------------------------------------------------------------
\subsection{Halo mass function}
-Being able to calculate the halo abundance as a function of mass is a necessary step to being able to constrain cosmology with probes such as galaxy clusters. While analytic functions are the traditional means of predicting evolution of halo abundances, calibration is frequently required against cosmological simulations. In order to reach the high precision required for cosmological constraints in a self-consistent fashion, it is ultimately necessary to use cosmological simulations; we implement this with halo mass functions with parameters fit to these simulations. The calculation of the halo mass function focuses around the spherical overdensity method of halo finding, in which the size of a halo can be defined with:
+Being able to calculate the halo abundance as a function of mass is a necessary step to being able to constrain cosmology with probes such as galaxy clusters. Modern cosmology makes extensive use of fitting functions in order to predict the evolution of halo abundances, which necessarily require derivation from cosmological simulations. In order to reach the high precision required for cosmological constraints in a self-consistent fashion, it is ultimately necessary to use cosmological simulations; we implement this with halo mass functions with parameters fit to these simulations. The calculation of the halo mass function focuses around the spherical overdensity method of halo finding, in which the size of a halo can be defined with:
\begin{equation}
\bar{\rho}(r_{\Delta}) = \Delta \times \bar{\rho}_{\mathrm{m}},
@@ -465,6 +465,7 @@ where the only change is in the formulation of the second term. Note that the fi
The \citet{Tinker2010} model parameterizes the halo mass function in terms of the peak height, $\nu = \delta_c/\sigma_M$, where $\delta_c=1.686$ is the critical density for collapse. The function is then re-expressed as
\begin{equation}
f(\nu) = \alpha[1+(\beta\nu)^{-2\phi}]\nu^{2\eta}e(-\gamma\nu^2/2).
+ \label{eq:tinkerf}
\end{equation}
We note that these halo mass functions, while implemented to high {\em numerical} accuracy through comparison against a flat $\Lambda$CDM cosmology (see `CCL1' in Table \ref{tab:cosmologies}), carry their own uncertainties. It has not been significantly studied whether the halo mass function is universal with respect to changes in dark energy parameterisation.
@@ -486,10 +487,9 @@ An important step in many interpretations of the halo model is to have a measure
As with measures of the halo mass function, high accuracy cosmological constraints requires the use of numerical simulations to develop fitting functions and emulators. We note that we will define haloes as in the above subsection focusing on the halo mass function. {\tt CCL} implements the halo bias fitting function results in \citet{Tinker2010}, though future improvements will likely require the use of emulator methods.
-The \citet{Tinker2010} model parameterizes the halo mass function and the halo bias in terms of the peak height and the critical density for collapse as
+The \citet{Tinker2010} model parameterizes the halo mass function (Eq.~\ref{eq:tinkerf}) and the halo bias in terms of the peak height and the critical density for collapse as
\begin{eqnarray}
b(\nu) &=& 1 - A\frac{\nu^a}{\nu^a + {\delta_c}^a} + B\nu^b+C\nu^c,\\
- f(\nu) &=& \alpha[1+(\beta\nu)^{-2\phi}]\nu^{2\eta}e(-\gamma\nu^2/2).
\end{eqnarray}
Again, while high {\em numerical} accuracy has been verified, there is a remaining uncertainty to the {\em physical accuracy}. \citet{Tinker2010} found a $\sim6\%$ scatter when determining the halo bias due to differences in simulations alone. In addition, this parameterization does not include a careful exploration of any impact due to changes in the dark energy equation of state. As with the halo mass function, studies will be required to reach accuracy at the percent level for any cosmological predictions. \elisa{I think we need some further references here of scale-dependen bias, assembly bias, etc.}
