 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-62/1/b/i/solution

Ω_{M} ↓ 0

, use

arsinh z = lo g (z + 1 + z^{2}) \sim lo g (2 z)

to obtain

H_{0} t_{0} \sim \frac{1}{3} lo g \frac{4}{Ω _{M}} ⟶ \infty.

At fixed Hubble constant, the age of a flat matter-Lambda universe diverges logarithmically. The limiting purely vacuum-dominated de Sitter spacetime has an exponential scale factor and no Big Bang at a finite value of cosmic time in this flat slicing.

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