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

Phantom energy with

w = - 4/3

has

ρ_{P} \propto a

, so flatness gives

\overset{a}{˙} = H_{0} Ω a 1 + β^{2} a, β^{2} = \frac{1 - Ω}{Ω} .

For

b^{2} = 1 + β^{2} a

, integration yields

lo g \frac{b - 1}{b + 1} = H_{0} Ω (t - t_{0}) + lo g κ, κ = \frac{1 - Ω}{1 + Ω} .

Solving gives the displayed scale factor

a (t) = \frac{4Ω e ^{H_{0} Ω (t - t_{0})}}{[ ( 1 + Ω ) - ( 1 - Ω ) e ^{H_{0} Ω (t - t_{0})} ] ^{2}} .

For

1 - Ω ≪ 1

and times near

t_{0}

, this approaches de Sitter growth

e^{H_{0} Ω (t - t_{0})}

. The denominator vanishes at

t_{BR} - t_{0} = \frac{1}{H _{0} Ω} lo g \frac{1 + Ω}{1 - Ω} .

Even a tiny phantom fraction therefore ends expansion in finite time; its divergent density ultimately disrupts loosely bound systems.

Solved by gpt-5.6-sol high.

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