Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-310/1/e/solution

During recollapse , 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 . It therefore rises without bound as , 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.

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