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\DocumentMetadata{
lang=en-US,
pdfversion=2.0,
pdfstandard=ua-2,
tagging=on,
tagging-setup={math/alt/use}
}
\documentclass[11pt]{article}
\usepackage[letterpaper,margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[american]{babel}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{enumext}
\hypersetup{pdftitle={Lecture note for Introduction to Cosmology}, pdfdisplaydoctitle}
\title{Lecture note for Introduction to Cosmology}
\author{Yao-Yuan Mao}
\date{February 6, 2026}
\begin{document}
\maketitle
\section*{Matter-Dominated, Flat Universe}
We begin with the Friedmann equation for a flat universe containing only (nonrelativistic) matter. There is only one term on the right-hand side:
\begin{equation}
\left( \frac{\dot a}{a} \right)^2 =
\frac{8\pi G}{3c^2}\, \frac{\epsilon_{m,0}}{a^3}.
\end{equation}
\section*{Quiz}
\begin{enumext}
\item Consider a \emph{non-flat} universe that contains matter, radiation, \emph{and} the cosmological constant ($\Lambda$). Put the following eras in chronological order.
\begin{enumext}[columns=2]
\item Curvature-dominated era
\item Matter-dominated era
\item Radiation-dominated era
\item $\Lambda$-dominated era
\end{enumext}
\end{enumext}
% \begin{figure}[h]
% \includegraphics[width=0.5\textwidth,clip,trim={0 1cm 0 1cm},alt={Repeating pattern of evenly spaced black dots arranged in staggered rows on a white background}]{dot_pattern.png}
% \end{figure}
\end{document}