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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-1/15d/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 15D / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

a^{2}

and differentiate:

\overset{a}{˙}^{2} + k c^{2} = \frac{8 π G}{3} ρ a^{2}, 2 \overset{a}{˙} \overset{a}{¨} = \frac{8 π G}{3} (\overset{ρ}{˙} a^{2} + 2 ρ a \overset{a}{˙}) .

(1)

Insert the cosmological continuity equation to obtain

2 \overset{a}{˙} \overset{a}{¨} = - \frac{8 π G}{3} a \overset{a}{˙} (ρ + 3 P / c^{2}) .

(2)

On a nonstatic branch divide by

2 a \overset{a}{˙}

; continuity extends the result across isolated turning points. The resulting Friedmann acceleration equation is

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

(3)

An everywhere static branch needs the independent spatial Einstein field equations: the two differentiated constraints alone cannot determine the pressure when

\overset{a}{˙} \equiv 0

. This is the qualification to deriving the acceleration equation solely by division.

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