Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / d

Table of contents
- Solution d

Solution

Cold dark matter has

c_{s} = 0

. During matter domination,

a (t) \propto t^{2/3}, H = \frac{2}{3 t}, \overset{ρ}{ˉ} = \frac{1}{6 π G t ^{2}} .

The density equation becomes

\ddot{δ} + \frac{4}{3 t} \dot{δ} - \frac{2}{3 t ^{2}} δ = 0.

Trying the power law

δ = t^{p}

gives

p (p - 1) + \frac{4}{3} p - \frac{2}{3} = 0,

whose roots are

p = 2/3

and

p = - 1

. Thus the cosmic-time matter density modes are

δ (t) = A t^{2/3} + B t^{- 1},

and the leading growing mode is

δ \propto t^{2/3} \propto a (t)

.

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