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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 1

For $k=-1$, the <Friedmann-Lemaître-Robertson-Walker metric> is
$$
ds^2=-dt^2+a^2(t)\left[d\chi^2+\sinh^2\chi
\,(d\theta^2+\sin^2\theta\,d\phi^2)\right].
$$
The empty first <Friedmann equation>, with $\Lambda=0$, gives
$$
\frac{\dot a^2-1}{a^2}=0.
$$
Choosing the expanding branch and shifting the origin of proper time yields
$$
\boxed{a(t)=t,\qquad t>0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D / a / Solution

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