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diff --git a/macros/latex/contrib/exam-n/sample/cosmo3.tex b/macros/latex/contrib/exam-n/sample/cosmo3.tex new file mode 100644 index 0000000000..b7b74c0107 --- /dev/null +++ b/macros/latex/contrib/exam-n/sample/cosmo3.tex @@ -0,0 +1,50 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Andrew Davies} + +The Friedmann equations are written, in a standard notation, +\begin{gather*} +H^2 = \frac{8\pi G\rho}{3}-\frac{kc^2}{R^2}+\frac{\Lambda}{3},\\ +\Diffl {}t (\rho c^2R^3)=-p\Diffl {R^3}t, +\end{gather*} +Discuss briefly the meaning of each of $H$, $\rho$ , $k$ and +$\Lambda$. \partmarks{4} + +Suppose the Universe consists of a single substance with equation +of state $p=w\rho c^2$, where $ w=$constant. Consider the +following cases, with $k=\Lambda=0$: + +\part For $w = 0$, find the relation between $R$ and $\rho$. Hence +show that $H=\frac{2}{3t}$. What is the physical interpretation of +this case? +\partmarks{8} + +\part In the case $w=-1$ , show that $H = $constant and $R = A \exp(Ht)$, +with $A$ constant. +\partmarks{4} + +\part Explain how the case, $w=-1$, $k=\Lambda=0$, $\rho=0$ is +equivalent to an empty, flat, Universe with a non-zero $\Lambda$. +\partmarks{2} + +\part Consider a model Universe which contained matter with equation +of state with $w = 0$ for $0 < t < t_0$, but which changes to +$W=0$ for $t\ge t_0$ without any discontinuity in $H(t)$. +Regarding this second stage as driven by a non-zero $\Lambda$ what +is the value of $\Lambda$ if $t_0 = 10^{24}$\units{\mu s}? Define the +dimensionless deceleration parameter, $q$, and find its value +before and after $t_0$. +\shout{Shout it loud: I'm a geek and I'm proud} +\partmarks{8} + +Note: that's +\[ +t_0=10^{24}\units{\mu s}\qquad\text{with a letter mu: $\mu$}. +\] + +\part To what extent does this idealized model resemble the currently +accepted picture of the development of our Universe? +\partmarks{4} +\end{question} +\end{document} |