% miko:type: atomic
% miko:title: Friedmann Equations
% miko:summary: The equations governing the expansion of a homogeneous and isotropic universe.
% miko:tags: cosmology
% miko:difficulty: undergrad
% miko:related: complete-cosmology, critical-density, cosmological-constant, scale-factor-redshift
% miko:keywords: FLRW, expansion, density, curvature

\documentclass{article}
\usepackage{amsmath}
\title{Friedmann Equations}
\begin{document}
\maketitle

\section{Setup}

Assume the universe is homogeneous and isotropic on large scales. The geometry
is described by the FLRW metric and matter is modeled as a perfect fluid with
density $\rho$ and pressure $p$.

\section{First Friedmann Equation}

The first Friedmann equation is

$$
H^2 =
\frac{8\pi G}{3}\rho
-\frac{kc^2}{a^2}
+\frac{\Lambda c^2}{3}.
$$

It relates the expansion rate to energy density, spatial curvature, and the
cosmological constant.

\section{Acceleration Equation}

The acceleration equation is

$$
\frac{\ddot a}{a}
=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)
+\frac{\Lambda c^2}{3}.
$$

Positive pressure gravitates in general relativity. Radiation therefore affects
expansion more strongly than nonrelativistic matter.

\section{Conservation}

The fluid equation is

$$
\dot\rho + 3H\left(\rho+\frac{p}{c^2}\right)=0.
$$

For an equation of state $p=w\rho c^2$, this gives $\rho\propto a^{-3(1+w)}$.

\end{document}
