Background Evolution

The Einstein equations $G_{\mu\nu} = 8\pi G T_{\mu\nu}$ express the metric evolution in terms of the matter sources. The background evolution equations are

$$\begin{array} {rcl}{\dot \rho_f \over \rho_f} + 3(1+w_f) {\d... ...dot a \over a} \dot\phi +a^2 {\cal V}_{,\phi} = 0\,,\end{array}$$

for the fluid and scalar field components respectively and

$$\begin{array} {rcl}\left( {\dot a \over a} \right)^2 + K = {8\pi G \over 3} a^2 ( \rho_f + \rho_\phi + \rho_v) \,,\end{array}$$ (36)

where $w_f=p_f/\rho_f$ and $\rho_\phi(\phi)$ was given in Eq. (17) and $\rho_v =3 H_0^2 \Omega_\Lambda / 8\pi G$ is the vacuum energy.