 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-66/1/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Write

H = \overset{a}{˙} / a

\frac{a ¨}{a} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c ^{2}}) + \frac{Λ c ^{2}}{3} .

Since

\dot{H} = \overset{a}{¨} / a - H^{2}

, eliminating

H^{2}

with the first equation yields

\dot{H} - \frac{k c ^{2}}{a ^{2}} = - 4 π G (ρ + \frac{p}{c ^{2}}) .

Differentiating the first equation, with constant cosmological constant and curvature parameter, now gives

\frac{8 π G}{3} \overset{ρ}{˙} = 2 H (\dot{H} - \frac{k c ^{2}}{a ^{2}}) = - 8 π G H (ρ + \frac{p}{c ^{2}}) .

\overset{ρ}{˙} + 3 H (ρ + p / c^{2}) = 0

. Multiplying by

a^{3}

and using

d (a^{3}) / d t = 3 H a^{3}

proves

\frac{d}{d t} (ρ a^{3}) + \frac{p}{c ^{2}} \frac{d}{d t} (a^{3}) = 0.

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