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\begin{center}
    {\large Astro 321: CMB Final Helper Problem Set 3}

\end{center}

\bigskip
\noindent{\large\bf Initial Conditions}
\smallskip

Take the initial conditions for a given $k$-mode to be $\zeta(\eta_i)=1$ where
$\eta_i$ is the initial time step for the integration (we will choose this
to be sufficiently early that the mode is well outside of the horizon and
the time is well before matter radiation equality).  We will scale this back
to an appropriate scale invariant spectrum later.  We will take a system of
CMB photons $\gamma$, baryons $b$ and cold dark matter $c$. Since we will ignore
neutrinos, the anisotropic stress $\pi$ is negligible in the initial conditions.

\begin{itemize}

\item{} From the relation
\begin{equation}
\Phi = {3 + 3 w \over 5 + 3 w} \zeta\,,
\end{equation}
$w=1/3$, the Poisson equation
\begin{equation}
k^2 \Phi = 4\pi G a^2 \sum_i \rho_i \Delta_i
\end{equation}
and the adiabatic condition
\begin{equation}
\Delta_c = \Delta_b = {3 \over 4}\Delta_\gamma
\end{equation}
find the initial conditions at $\eta_i$ for the three comoving density perturbations.


\item{} Again with the adiabatic condition $v_c=v_b=v_\gamma$ and the
continuity equation for $\dot \Delta$ derived in PS 5. Find the initial
conditions for the velocities at $\eta_i$.  

\item{} Since the anisotropic stress vanishes, the initial conditions for
the Newtonian potential $\Psi = -\Phi$.

\end{itemize}

These are the IC's for your Runge-Kutta integration.
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