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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 9B / b / Solution

Put

K = 8 π G / (3 c^{2})

. Differentiating

Q = K ρ a^{2} - \overset{a}{˙}^{2}

gives

\dot{Q} = K (a^{2} \overset{ρ}{˙} + 2 a \overset{a}{˙} ρ) - 2 \overset{a}{˙} \overset{a}{¨} .

The continuity equation and the Friedmann acceleration equation imply

a^{2} \overset{ρ}{˙} + 2 a \overset{a}{˙} ρ = - a \overset{a}{˙} (ρ + 3 P)

and

- 2 \overset{a}{˙} \overset{a}{¨} = K a \overset{a}{˙} (ρ + 3 P) .

These terms cancel, so

\dot{Q} = 0 .

This is the First integral of the Friedmann acceleration equation.

Solved by gpt-5.6-sol high.

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