updated with halo bias section
@@ -371,11 +371,6 @@ f(\sigma)=A\Big[\Big(\frac{b}{\sigma}+1\Big)^{-a}\Big]e^{-c/{\sigma}^2},
\end{equation}
where the only change is in the formulation of the second term. Note that the fitting parameters in the \citet{Angulo2012} formulation do not contain any redshift dependence and the use of it is primarily for testing and benchmark purposes.
-%The \citet{Tinker2010} model parameterizes both the halo mass function and the halo bias in terms of the peak height, $\nu = \delta_c / \sigma(M)$, where $\delta_c$ is the critical density for collapse and is chosen to be $1.686$ for this particular parameterization. We can then parameterize the halo function and halo bias 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}
The \citet{Tinker2010} model parameterizes the halo mass function in terms of th peak height, $\nu = \delta_c/\sigma(M)$, where $\delta_c=1.686$ is the critical density for collapse. The multiplicity function is then written as
\begin{equation}
f(\nu) = \alpha[1+(\beta\nu)^{-2\phi}]\nu^{2\eta}e(-\gamma\nu^2/2).
@@ -387,6 +382,21 @@ We note that these halo mass functions, while implemented to high {\em numerical
\citet{Tinker2008,Tinker2010} quote 5\% accuracy of their mass funcitons.
This result is consistent with the work of \citet{Watson2013}, which also finds a $5\%$ level difference in comparison to the Tinker fitting function. Further study will be required in the future in order to gain percent level accuracy in determining the halo mass function.
+\subsection{Halo bias}
+\vol{Antonio Villarreal}
+
+\cont{Halo bias definition.}
+
+An important step in many interpretations of the halo model is to have a measure of the bias of dark matter halos in contrast to the underlying density field. 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} currently 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 bias in terms of the peak height, $\nu = \delta_c / \sigma(M)$, where $\delta_c$ is the critical density for collapse and is chosen to be $1.686$ for this particular parameterization. We can then parameterize the halo function and halo bias 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. \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.
+
%-------------------------------------------------------------------------------
\subsection{Massive neutrinos}
\vol{Danielle Leonard}
