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{\large Astro 321: Inflation 1}
Due Feb. 6
\end{center}
\section{\large Problem 1: ``Gauge Transformations'' and the Bardeen-$\zeta$}
Recall from class the general equations of motion in relativistic
linear perturbation theory. [A flat universe is assumed throughout. Dots
are conformal time derivatives and $w \equiv p/\rho$.]
\smallskip
\noindent
The continuity/energy equation:
\begin{equation}
\dot \delta = -3 {\dot a \over a} \left( c_s^2 - w \right) \delta
- (1+w)(kv + 3\dot\Phi)\,,
\end{equation}
with $c_s^2 \equiv \delta p/\delta \rho$, where the fluctuation
$\delta p$ is not to be confused
with $\delta \times p$.
The Euler equation:
\begin{equation}
\dot v = -(1-3w){\dot a \over a} v - {\dot w \over 1+w} v
+ {kc_s^2 \over 1+w} \delta - {2 \over 3}{w \over 1+w} k\pi
+ k\Psi\,.
\end{equation}
The Poisson equation:
\begin{eqnarray}
k^2 \Phi &=& 4\pi G a^2 \rho [ \delta + 3 {\dot a \over a} (1+w) v/k] \,,\nonumber\\
k^2 (\Psi+\Phi) & = & -8\pi Ga^2 p\pi\,,
\end{eqnarray}
and an redunant combo of these equations that you will find useful:
\begin{equation}
\left( \dot a \over a \right) \Psi - \dot \Phi = 4\pi G a^2 (\rho+p) v/k\,.
\label{eqn:aux}
\end{equation}
Although the representation of the system in these variables [called
the ``Newtonian gauge'' or ``longitudinal guage'' system] is complete and
best corresponds to our Newtonian intuition, it is inconvenient for
both numerical and analytic work on certain problems. In particular the
gravitational potentials $\Phi$
and $\Psi$ and their time evolution are not simply related to
the matter fields.
Let look for a more convenient representation. You may
think of this operation as purely a change of variables but for the
GR cognescenti, this operation is a gauge transformation (of a time
shift $v/k$) and the
transformed matter fluctuation fields are simply the matter fluctuation
fields in a ``comoving'' set of coordinates.
\smallskip
Noting the form of the Poisson equation, define a new density perturbation
\begin{equation}
\Delta \rho = \delta \rho - \dot \rho v/k
\end{equation}
\begin{itemize}
\item Show
\begin{equation}
\Delta \equiv \delta + 3{\dot a \over a}(1+w) v/k
\end{equation}
\item
Rewrite the continuity equation and show that
\begin{equation}
\dot \Delta = -3 {\dot a \over a} \left( C_s^2 - w \right) \Delta
- (1+w)(kv + 3\dot\zeta)\,,
\end{equation}
where the transformed sound speed
\begin{eqnarray}
C_s^2 &\equiv& {\Delta p \over \Delta \rho}\\
\Delta p &\equiv & \delta p - \dot p v/k
\end{eqnarray}
(again don't confuse $\Delta p$ with $\Delta \times p$),
and the Bardeen curvature $\zeta$
\begin{equation}
\zeta \equiv \Phi - {\dot a \over a} v/k\,.
\end{equation}
Now the potential is defined simply in terms of the matter fields
$k^2 \Phi = 4\pi G a^2 \rho \Delta$ as you would expect from Newtonian
gravity but at the price of introducing $\dot\zeta$ into its evolution
equation.
The introduction of $\dot \zeta$ is in fact also useful in that it
is also simply related to the matter fields.
\item
Show that
\begin{eqnarray}
\dot \zeta & = & {\dot a \over a}\xi \nonumber\\
\xi & = & - {C_s^2 \over 1+w} \Delta + {2 \over 3} {w \over 1+w} \pi\,.
\end{eqnarray}
Hint: differentiate the definition of $\zeta$ and use the Euler equation to
eliminate terms. You may find the auxiliary equation (\ref{eqn:aux}) and
the acceleration equation for $\ddot a$ useful.
Since $\xi$ is then directly related to the stress fluctuations, what this
says is that if stress fluctuations can be ignored, as they can always
be outside the horizon, the variable $\zeta$ is a constant, {\it independently}
of the nature of the matter fields. This enormously useful fact proven
first by Bardeen (1980) allows us to ignore the details of many processes
since once $\zeta$ is calculated you are done with perturbation theory
on large scales!
For the CMB final project: this representation of the linear perturbation
equations is numerically stable, unlike the original form.
\end{itemize}
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