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\newcommand{\Titleshort}{MHD through recombination}
\newcommand{\Author}{Alberto Roper Pol}
\newcommand{\Title}{\Titleshort}
\newcommand{\AuthorName}{Alberto Roper Pol}
\newcommand{\Date}{\today}

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\begin{document}
\maketitle

\section{MHD equations}

\subsection{Continuity equation}
\begin{equation}
    \boxed{
    \partial_\tau \tilde T^{00} + \partial_i \tilde T^{0i} + \frac{a'}{a}
    \tilde \rho \, (3w - 1) = 0.}
\end{equation}

\begin{align}
    \biggl[1 - \frac{w}{(1 + w) \gamma^2}
    \biggr]
    \partial_\tau \ln \tilde \rho &\, +  \bigl(1 - \gamma^{-2} \bigr) \partial_\tau \ln (1 + w) + \partial_\tau \ln \gamma^2 \nonumber \\
    &\, = - \nab \cdot \uu
    - \uu \cdot \nab \ln \bigl(\tilde \rho \gamma^2 \bigr) + \frac{\tilde f^0}{(1 + w) \tilde \rho \gamma^2} - \frac{a'}{a}
    \frac{3w - 1}
    {(1 + w) \, \gamma^2}.
\end{align}

Subrelativistic regime $\gamma^2 \ll 1$,
\begin{align}
    \partial_\tau \ln \tilde \rho = - (1 + w) \Bigl[ \nab \cdot \uu
    + \uu \cdot \nab \ln \tilde \rho \Bigr]  + \frac{\tilde f^0}{\tilde \rho} - \frac{a'}{a}
    (3w - 1).
    \label{cont_sub}
\end{align}

\subsection{Momentum equation}

\begin{equation}
    \boxed{\partial_\tau \tilde T^{0i} 
    + \partial_j \tilde T^{ij} = 0.}
\end{equation}

\begin{align}
    \partial_\tau \uu + \uu \Bigl[ \partial_\tau \ln \tilde \rho \,
    + &\, \partial_\tau \ln (1 + w) + \partial_\tau \ln \gamma^2 + 
    \bigl(\uu \cdot \nab\bigr)
    \ln \bigl( \tilde \rho \gamma^2\bigr) + \nab \cdot \uu \Bigr] \nonumber \\
    &\, = - \bigl(\uu \cdot \nab \bigr) \uu - \frac{w}{1 + w} \frac{\nab \ln \tilde \rho}{\gamma^2} + \frac{\tff}{\tilde \rho\, (1 + w) \gamma^2}.
    \label{mom_1}
\end{align}

Subrelativistic regime $\gamma^2 \ll 1$,
\begin{align}
    \partial_\tau \uu + \uu \Bigl[ \partial_\tau \ln \tilde \rho \,
    + \partial_\tau \ln (1 + w) + 
    \bigl(\uu \cdot \nab\bigr)
    \ln \tilde \rho + \nab \cdot \uu \Bigr]
    = - \bigl(\uu \cdot \nab \bigr) \uu - \frac{w}{1 + w} \nab \ln \tilde \rho + \frac{\tff}{\tilde \rho\, (1 + w)}.
    \label{mom_1}
\end{align}
Introducing \Eq{cont_sub}, one gets
\begin{align}
    \partial_\tau \uu = &\, - \uu \cdot \nab \uu  + w \uu
    \Bigl(\uu \cdot \nab \ln \tilde \rho + \nab \cdot \uu
    \Bigr)
     - \frac{w}{1 + w} \nab \ln \tilde \rho + 
     \frac{\tff}{\tilde \rho\, (1 + w)} \nonumber \\
     &\, - \uu \biggl[
    \frac{\tilde f_0}{\tilde \rho} - \frac{a'}{a} (3w - 1) + \partial_\tau \ln (1 + w) + \alpha \biggr].
    \label{mom_2}
\end{align}

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\end{document}
