Expansion of the Universe

The Friedmann-Walker-Robertson metric is

The scale factor evolves according to

\[\left(\frac{\dot a}{H_0}\right) \pm \left(1 - \Omega_0 + \frac{\Omega_M}{a} + \frac{\Omega_R}{a^2} + a^2\Omega_\Lambda\right)^{1/2}.\]