The <constant-equation-of-state density scaling> gives
$$
\bar\rho_D\propto a^{-3(1+w)}=a^{-1}.
$$
For a spatially flat universe dominated by this component, the <Friedmann equation> gives $H^2\propto a^{-1}$. Integrating $\dot a/a\propto a^{-1/2}$ yields
$$
\boxed{a(t)\propto(t-t_*)^2,
\qquad H(t)=\frac{2}{t-t_*}}.
$$
This is accelerated expansion because $w=-2/3<-1/3$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / e / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-310/4/e/i/solution