% miko:type: atomic
% miko:title: Critical Density
% miko:summary: The density that makes the spatial curvature term vanish for a given Hubble rate.
% miko:tags: cosmology
% miko:difficulty: undergrad
% miko:related: complete-cosmology, friedmann-equations, cosmological-constant
% miko:keywords: density parameter, flat universe, Omega

\documentclass{article}
\usepackage{amsmath}
\title{Critical Density}
\begin{document}
\maketitle

\section{Definition}

The critical density is

$$
\rho_c = \frac{3H^2}{8\pi G}.
$$

It is the density required for a spatially flat universe in the simplest
Friedmann equation.

\section{Density Parameters}

Define

$$
\Omega_i = \frac{\rho_i}{\rho_c}.
$$

The Friedmann equation can then be written as a relation among $\Omega$ values,
which is convenient for comparing matter, radiation, curvature, and dark
energy contributions.

\end{document}
