added problems from structure formation.

tutorials/cosmology.tex

@@ -67,3 +67,70 @@ Explain the physical origin of the differences.
 \end{enumerate}
 \end{enumerate}
 \end{enumerate}
 \end{enumerate}
 
 
+
+\subsection{Linear Structure Formation}
+
+We work in the Newtonian regime on sub-horizon scales.
+Unless stated otherwise, assume a homogeneous background described by scale factor
+$a(t)$ and Hubble parameter $H = \dot a/a$. In comoving coordinates, the Newtonian fluid equations are:
+
+\begin{eqnarray}
+	{\dot \delta} + \frac{1}{a}\nabla \cdot \bm{v} = 0 ~~ \mathrm{(Continuity)} \\
+	{ \bm{\dot v}} + H\bm{v} = -\frac{1}{a}\nabla \phi - \frac{1}{a\bar\rho}\nabla p ~~ ~~ \mathrm{(Euler)} \\
+	\nabla^2 \phi = 4\pi G a^2 \bar\rho\, \delta ~~\mathrm{(Poisson)}. 
+\end{eqnarray}
+Here,  $ \delta \equiv ({\rho - \bar\rho})/{\bar\rho}$ is the density contrast and $\bm{v}$ is the peculiar velocity.
+
+
+\subsubsection{Problems}
+
+\begin{enumerate}
+
+\item Assume $|\delta| \ll 1$ and linearize the equations. Take the divergence of the Euler equation and use the continuity
+equation to eliminate $\nabla\cdot\bm{v}$. Using the Poisson equation to eliminate $\phi$, show that
+
+\begin{eqnarray}
+\ddot\delta + 2H\dot\delta
+- 4\pi G \bar\rho\, \delta - \frac{c_s^2}{a^2}\nabla^2 \delta = 0,
+\end{eqnarray}
+where $c_s^2 = \partial p/\partial\rho$. What physical timescales appear in this equation? When does instability occur? 
+
+\item Assume pressureless dark matter ($c_s = 0$). In a matter-dominated universe:
+$a(t) \propto t^{2/3}, ~~ H = \frac{2}{3t}, ~~ \bar\rho = \frac{1}{6\pi G t^2}.$ Substitute these expressions into the growth equation and show that
+\begin{equation}
+\ddot\delta + \frac{4}{3t}\dot\delta
+- \frac{2}{3t^2}\delta = 0.
+\end{equation}
+
+\item Assume a power-law solution $\delta \propto t^n$ and solve for $n$.
+Show that the growing mode scales as
+\begin{equation}
+\delta \propto t^{2/3} \propto a(t).
+\end{equation}
+
+\item Why is the growth only linear in $a(t)$ rather than exponential?
+Discuss the role of the Hubble drag term.
+
+\item Now include pressure. Fourier-decompose the perturbation: $\delta(\bm{x},t) = \delta_k(t)e^{i\bm{k}\cdot\bm{x}}.$ Show that each mode satisfies
+
+\begin{equation}
+\ddot\delta_k + 2H\dot\delta_k
++ \left(\frac{c_s^2 k^2}{a^2}
+- 4\pi G \bar\rho\right)\delta_k = 0.
+\end{equation}
+
+\item Discuss the behavior in the limits: 1) $k \rightarrow 0$ (large scales), 2) large $k$ (small scales). Explain physically when perturbations grow and when they oscillate.
+
+\item Consider a static background ($H=0$). Define the Jeans wavenumber $k_J$ and show that $k_J^2 = \frac{4\pi G\bar\rho}{c_s^2}$.
+
+\item Define the Jeans length $\lambda_J = 2\pi/k_J$.
+Show that the Jeans mass scales as $M_J \sim \frac{c_s^3}{G^{3/2}\bar\rho^{1/2}}.$
+
+\item In an expanding universe, is the Jeans scale constant?
+Discuss qualitatively how expansion modifies gravitational instability.
+
+\item Show that instability occurs when the free-fall time $t_{\rm dyn} \sim (G\bar\rho)^{-1/2} $ is shorter than 1)  the sound-crossing time and 2) the expansion time $H^{-1}$.  Discuss how structure formation can be understood entirely
+as a competition between timescales (in the spirit of our first lecture).
+\end{enumerate}
+
+