Friedmann Equations

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 $p$.

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.

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.

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 $\rho\propto a^{-3(1+w)}$.