FRW universe models

In a homogeneous and isotropic universe the evolution of the scale factor $a(t)$ is governed by the Friedmann equation. With the present-day density parameters of radiation, matter, curvature and the cosmological constant it reads

$$H^2(a) = \left(\frac{\dot a}{a}\right)^2 = H_0^2 \left( \Omega_{\mathrm{r}}\, a^{-4} + \Omega_{\mathrm{m}}\, a^{-3} + \Omega_k\, a^{-2} + \Omega_\Lambda \right),$$

and the age of the universe follows from integrating $dt = da / (a H)$ from the Big Bang to today:

$$t_0 = \int_0^1 \frac{da}{a\,H(a)}.$$

For a general mix of components the age integral cannot be evaluated analytically, so one has to use numerical methods. The video below walks through the resulting universe models.