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{\large Astro 321: Problem Set 3}
Due Jan. 30
\end{center}
This problem set will begin a series which builds your standard toolkit
for structure formation calculations. The motivation of some pieces
may seem mysterious for now, but save the codes you build here because
you will need them again later in the course.
\section{\large Problem 1: Power Spectra}
\begin{itemize}
\item Define the power spectrum today as
\begin{eqnarray}
\Delta^2(k) &\equiv& {k^3 \over 2\pi^2} P(k) \nonumber\\
&=& \delta_H^2 { \left( k \over H_0 \right) }^{3+n} T^2(k)
\end{eqnarray}
For reference, $\delta_H$ is the normalization of density perturbations
on the horizon scale today $k \approx H_0$:
\begin{eqnarray}
\delta_H = 1.94 \times 10^{-5} \Omega_m^{-0.785-0.05 \ln \Omega_m}
\exp[-0.95(n-1) - 0.169(n-1)^2]
\end{eqnarray}
is the famous COBE normalization courtesy of Bunn \& White,
$n$ is the spectral index
of the {\it initial} density fluctuations and $T(k)$ is the {\it transfer
function} which defines the linear response (through
gravitational perturbation theory) to the initial perturbations.
We will see later where that comes from; for now take it to be defined
as
\begin{eqnarray}
T(k(q)) &=& {L(q) \over L(q) + C(q) q^2} \nonumber\\
L(q) &=& \ln(e + 1.84 q) \nonumber\\
C(q) &=& 14.4 + {325 \over 1+ 60.5 q^{1.11}}
\end{eqnarray}
$q$ scales $k$ to the horizon size at matter radiation equality and is given
for historical reasons by
\begin{eqnarray}
q = {k \over \Omega_m h^2 {\rm Mpc}^{-1}} (T_{\rm CMB}/2.7K)^2 \qquad
T_{\rm CMB}=2.728K
\end{eqnarray}
\item Calculate $\eta(a_{\rm eq})$ and show that $k\eta(a_{\rm eq})$
has the above scaling, $q \propto k\eta(a_{\rm eq})$ in its scalings
with $\Omega_m$, $h$, $T_{\rm CMB}$.
For what value of $q$ is $k\eta(a_{\rm eq})=1$?
At this value of $q$, what is the value of the transfer function $T$?
\item Write a code in the language of your choice
to generate $\Delta^2(k)$ with $k$ in units of
$h$ Mpc$^{-1}$. Leave $\Omega_m$, $n$, $h$ as adjustable parameters.
Notice that value of $q$ with $k$
in $h$ Mpc$^{-1}$ depends on $\Omega_m h$, which is usually
called the shape parameter ``$\Gamma$'' in the literature.
\item For an $\Omega_m=1/3$, $h=1/\sqrt{2}$, $n=1$ cosmology,
plot your result of $\Delta^2(k)$. For what $k$ in
$h$ Mpc$^{-1}$ is $\Delta^2=1$.
\end{itemize}
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