Added caveat about Gadget's time-integration operators for entropy

theory/Cosmology/operators.tex

@@ -30,12 +30,18 @@ eq.~\ref{eq:cosmo_eom_v}) are integrated forward in time using $\Delta
 t_{\rm kick,h}$, whilst the accelerations given by the gravity forces
 (\nth{3} term in eq.~\ref{eq:cosmo_eom_v}) use $\Delta t_{\rm
   kick,g}$. The entropy or internal energy is integrated forward in
-time using $\Delta t_{\rm kick,A} = \Delta t_{\rm drift}$, whilst the
-change in energy due to the expansion of the Universe (first term in
-.q.~\ref{eq:cosmo_eom_u}) can be computed using
+time using $\Delta t_{\rm kick,A} = \Delta t_{\rm
+  drift}$\footnote{Note that {\sc gadget-2} uses a slightly different
+  operator here. They first multiply $\dot{A}_i'$ by $1/H$ and do not
+  not consider the $1/a^2$ term as part of the time-integration
+  operator. The $1/H$ term then integrates out trivially. This slight
+  inconsistency with the rest of the time-integration operators is
+  unlikely to lead to any practical difference.}, whilst the change in
+energy due to the expansion of the Universe (first term in
+eq.~\ref{eq:cosmo_eom_u}) can be computed using
 \begin{equation}
   \int_{a_n}^{a_{n+1}} H dt = \int_{a_n}^{a_{n+1}} \frac{da}{a} =
-  \log{a_{n+1}} - \log{a_n}
+  \log{a_{n+1}} - \log{a_n}.
 \end{equation}
 Following the same method as for the age of the Universe
 (sec. \ref{ssec:flrw}), the three non-trivial integrals are evaluated