5  Energy Components of the Universe with Time

Let us pick up again with the Friedmann equation \[ \left(\frac{\dot{a}}{a}\right)^{2} = H^2 = \frac{8 \pi G}{3} \rho(t) - \frac{k}{R_0^{2} a^{2}} \] To make progress, we need to know \(\rho\) as a function of \(a\). We can get this from the continuity equation \[ \dot{\rho} + 3 H (\rho + P) = 0 \] We need to specify what \(P\) is - we do so via an equation of state \[ P = w \rho \] where \(w\) need not be constant (but is for many single components of the Universe). Substituting into the continuity equation, we have \[ \dot{\rho} = - 3 \frac{\dot{a}}{a} (1+w(a)) \rho \] or \[ \rho = \rho_{0} \exp \left( -3 \int_{0}^{\ln a} (1+ w(a)) \, d\ln a \right) \] Now, standard energy components have constant \(w\)

\[ w=0 \implies \rho = \rho_{0}a^{-3} \]