added cosmology problems.

tutorials/Lab.tex

@@ -23,6 +23,9 @@
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+\textwidth 6.7in 
+\textheight 9.9in
+
 \begin{document}
 \begin{document}
 
 
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 \section{Astrophysics across scales}
 \section{Astrophysics across scales}
 \input{astro_scale.tex}
 \input{astro_scale.tex}
 
 
-% 
+\section{Cosmology}
+\input{cosmology.tex}
 
 
 \bibliography{Lab}
 \bibliography{Lab}
 \end{document}
 \end{document}

tutorials/cosmology.tex

+
+\subsection{Background cosmology}
+
+The Friedmann-Robertson-Walker (FRW) metric is
+\begin{equation}
+ds^2 = -c^2 dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2 \, d\Omega^2 \right]. 
+\end{equation}
+Above, $d\Omega^2 = d\theta^2 + \sin^2\theta\, d\phi^2$, and $r, \theta, \phi$ are comoving coordinates, while $k = 0$ for spacially flat universe and $k \pm 1$ for positively/negatively curved universe. We defined the comoving distance $\chi$ and proper distance $D_p$ and showed that, for a spatially flat universe, $D_p(t) = a(t)\, \chi$.
+
+\subsubsection{Problems:}
+
+\begin{enumerate}
+\item Consider a radially propagating light ray. Derive an expression for the comoving
+distance to a source observed at redshift $z$.
+
+\item Explain qualitatively how curvature affects the comoving volume element.
+What are the consequences for number counts of quasars as a function of redshift? 
+Can this be used as a way to constrain the spatial curvature of the universe? 
+What assumptions are required for this? 
+
+\item Define the angular diameter distance $D_A$ and the luminosity distance $D_L$.
+Show that in any FRW universe,
+\[
+D_L = (1+z)^2 D_A.
+\]
+Explain physically why these distances differ.
+
+\item How does spatial curvature modify the luminosity distance at fixed redshift?
+What observable effects would this have for standard candles and standard rulers?
+
+\item Consider a test particle of mass $m$ at the edge of a uniform spherical region
+of radius $R(t)$ and density $\rho(t)$.
+
+\begin{enumerate}
+\item Write down the Newtonian equation of motion for $R(t)$.
+
+\item Show that the total energy per unit mass may be written as
+\[
+E = \frac{1}{2}\dot R^2 - \frac{GM}{R},
+\]
+where $M = \frac{4\pi}{3}\rho R^3$.
+Interpret this equation as an energy balance and connect it to the virial theorem.
+
+\item Define the scale factor by $R(t) = a(t)r$, and show that the energy equation
+can be rewritten in the form
+\[
+\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}.
+\]
+This is the Newtonian analog of the Friedmann's equations. 
+\end{enumerate}
+
+\item The luminosity distance in an expanding universe is given by
+\[
+D_L(z) = (1+z)c \int_0^z \frac{dz'}{H(z')}.
+\]
+
+\begin{enumerate}
+\item Plot $D_L(z)$ for $0 \le z \le 3$ for the following cosmological models: (1) Einstein–de Sitter ($\Omega_m=1, \Omega_\Lambda=0$), (2)  Flat $\Lambda$CDM ($\Omega_m=0.3, \Omega_\Lambda=0.7$), (3) Open universe ($\Omega_m=0.3, \Omega_\Lambda=0$)
+
+\item At what redshift do the models begin to diverge significantly?
+Explain the physical origin of the differences.
+
+\item Here~\cite{sndata} you are given a dataset from the Supernova Cosmology project~\cite{SNCosmology}. This contains the redshift $z$, the distance modulus $\mu$, and the error on the distance modulus $\delta \mu$ measured from several Type 1a supernovae. The distance modulus is related to the luminosity distance (in parsecs) by $\mu = 5 \left(\log_{10} D_L - 1 \right)$. Use the supernova data from the entire redshift to estimate $H_0$ and $\Omega_M$. You can use SciPy's \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html}{\texttt{curve\_fit}} function to fit the data. 
+
+
+\end{enumerate}
+\end{enumerate}
+