addressed comment on fitting function f and comment on rho_m
@@ -426,14 +426,14 @@ Being able to calculate the halo abundance as a function of mass is a necessary
\bar{\rho}(r_{\Delta}) = \Delta \times \bar{\rho}_{\mathrm{m}},
\end{equation}
%
-where a spherical halo with radius $r_{\Delta}$ has an average density $\bar{\rho}$ equal to the overdensity parameter $\Delta$ times the mean background density of the universe at a given redshift, $\bar\rho_{\mathrm{m}}(z)$. \todo{I believe we use the same quantity without bars above. Make consistent? - DL} Within the literature, the choice of $\Delta$ can vary considerably, as observations focusing on the compact cores of halos often take much larger values of $\Delta$ than the fiducial definition in most halo clustering studies, $\Delta = 200$. We note that another common definition exists which utilizes the critical density of the universe, $\rho_{\mathrm{crit}}$; this introduces a simple conversion factor between the two definitions that must be accounted for. Currently, \ccl only accepts overdensity parameters with respect to the mean matter density, but we plan to allow for self-consistent handling of critical density based definitions in the future.
+where a spherical halo with radius $r_{\Delta}$ has an average density $\bar{\rho}$ equal to the overdensity parameter $\Delta$ times the mean background density of the universe at a given redshift, $\bar\rho_{\mathrm{m}}(z)$. \todo{I believe we use the same quantity without bars above. Make consistent? - DL}\asv{I would prefer the usage with bars throughout, as it is more explicit and matches uses in the simulation literature. Either direction I believe works though.} Within the literature, the choice of $\Delta$ can vary considerably, as observations focusing on the compact cores of halos often take much larger values of $\Delta$ than the fiducial definition in most halo clustering studies, $\Delta = 200$. We note that another common definition exists which utilizes the critical density of the universe, $\rho_{\mathrm{crit}}$; this introduces a simple conversion factor between the two definitions that must be accounted for. Currently, \ccl only accepts overdensity parameters with respect to the mean matter density, but we plan to allow for self-consistent handling of critical density based definitions in the future.
The halo mass function is defined as
\begin{equation}
\frac{dn}{dM}=f(\sigma)\frac{\bar{\rho}_\mathrm{m}}{M}\frac{d\ln{\sigma^{-1}}}{dM},
\label{eq:halo_mass_function}
\end{equation}
-where $n$ is the number density of halos of a given mass $M$ associated with the RMS variance of the matter density field $\sigma^2$ at a given redshift and $f$ is the derived fitting function. \todo{Clarify that this should not be confused with $f$ the linear growth rate of structure as introduced above? - DL}
+where $n$ is the number density of halos of a given mass $M$ associated with the RMS variance of the matter density field $\sigma^2$ at a given redshift and $f$ is the derived fitting function\footnote{Not to be confused with the linear growth rate of structure!}. \todo{Clarify that this should not be confused with $f$ the linear growth rate of structure as introduced above? - DL}
In \ccl calling this function returns the mass function in logarithmic mass bins, $dn/d\log_{10}{M}$, where the input is the halo mass $M$ and scale factor $a$.
The halo mass $M$ is related to $\sigma$ by first computing the radius $R$ that would enclose a mass $M$ in a homogeneous Universe at $z=0$:
