Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3 / c

Table of contents
- Solution c

Solution

Use

d / d t = a H, d / d a

. Then

\dot{δ}_{m} = a H \frac{d δ _{m}}{d a},

\ddot{δ}_{m} = a^{2} H^{2} \frac{d ^{2} δ _{m}}{d a ^{2}} + a H (H + a H^{'}) \frac{d δ _{m}}{d a},

where now

H^{'} = d H / d a

. Also

4 π G \overset{ρ}{ˉ}_{m} = \frac{3}{2} Ω_{m, 0} H_{0}^{2} a^{- 3} .

Substitution into part (a) and division by

a^{2} H^{2}

gives the matter growth equation as a function of scale factor

\frac{d ^{2} δ _{m}}{d a ^{2}} + (\frac{d lo g H}{d a} + \frac{3}{a}) \frac{d δ _{m}}{d a} - \frac{3 Ω _{m, 0} H _{0}^{2}}{2 a ^{5} H ^{2}} δ_{m} = 0 .

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