Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / c / Solution

With zero curvature and no cosmological constant, the present normalization requires ${\Omega }_M=1$. Thus $\dot{a}=H_0{a}^{-1/2}$, and choosing the big bang as $t=0$ givesFor an empty expanding model, ${\Omega }_k=1$ and the curvature must be negative. The equation becomes $\dot{a}=H_0$, givingThe latter is the Milne slicing of flat spacetime; its zero-scale-factor origin is not a matter-density curvature singularity. In the matter model, gravity decelerates expansion, so the earlier slope was greater than the present slope and the same present size was reached in less time. With the stipulated $H_0$, the two model ages are approximately $8.69$ and $13.0$ Gyr.