Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / e / Solution

During recollapse $H<0$, so nearby comoving galaxies are increasingly blueshifted rather than redshifted. Their physical separations and angular-diameter distances shrink, making them appear larger and generally brighter; in a closed geometry sufficiently old light can also produce repeated or strongly focused images.

The cosmic microwave background temperature obeys $T_{\gamma }\propto a^{-1}$. It therefore rises without bound as $\theta \to 2\pi$, and its photons are blueshifted. Before the formal Big Crunch, the growing temperature reionizes matter, scattering makes the universe opaque, and the idealization that old galaxies and the original last-scattering surface remain directly visible eventually fails.