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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 1

<Phantom energy> with $w=-4/3$ has $\rho_P\propto a$, so flatness gives
$$
\dot a=H_0\sqrt\Omega\,a\sqrt{1+\beta^2a},\qquad
\beta^2=\frac{1-\Omega}{\Omega}.
$$
For $b^2=1+\beta^2a$, integration yields
$$
\log\frac{b-1}{b+1}=H_0\sqrt\Omega(t-t_0)+\log\kappa,\qquad
\kappa=\frac{1-\sqrt\Omega}{1+\sqrt\Omega}.
$$
Solving gives the displayed scale factor
$$
a(t)=\frac{4\Omega e^{H_0\sqrt\Omega(t-t_0)}}
{\left[(1+\sqrt\Omega)-(1-\sqrt\Omega)e^{H_0\sqrt\Omega(t-t_0)}\right]^2}.
$$
For $1-\Omega\ll1$ and times near $t_0$, this approaches de Sitter growth $e^{H_0\sqrt\Omega(t-t_0)}$. The denominator vanishes at
$$
t_{\rm BR}-t_0=\frac1{H_0\sqrt\Omega}
\log\frac{1+\sqrt\Omega}{1-\sqrt\Omega}.
$$
Even a tiny phantom fraction therefore ends expansion in finite time; its divergent density ultimately disrupts loosely bound systems.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 15D / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 15D / c / Solution

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