Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 9A / Solution

The constant $R$ is the curvature radius of the closed spatial slices when $a=1$; at general scale factor their curvature radius is $Ra$ and their scalar curvature is $6/(R^2{a}^2)$.

The cosmological perfect-fluid continuity equation isDuring expansion, $\dot{a}>0$, and thereforeIf $C=\rho (t_0)a(t_0{)}^3$, then throughout the later expanding phaseMultiplying the Friedmann equation by $a^2$ givesThe right-hand side is negative oncewhich is impossible because ${\dot{a}}^2\ge 0$. Thus the scale factor cannot grow without bound.

In fact the expanding phase ends in finite time. Combining the Friedmann and continuity equations yields the Friedmann acceleration equationThe curvature term in the first Friedmann equation also impliesWhile $a$ remains below the finite bound just found, $\ddot{a}$ is bounded above by a strictly negative constant. Hence $\dot{a}$ reaches zero after finite time: the closed universe attains a maximum size and cannot expand forever. This is recollapse of a closed Friedmann universe with nonnegative pressure.

Solved by gpt-5.6-sol high.