\overset{ρ}{ˉ}_{D} \propto a^{- 3 (1 + w)} = a^{- 1} .

For a spatially flat universe dominated by this component, the Friedmann equation gives

H^{2} \propto a^{- 1}

. Integrating

\overset{a}{˙} / a \propto a^{- 1/2}

yields

a (t) \propto (t - t_{*})^{2}, H (t) = \frac{2}{t - t _{*}} .

This is accelerated expansion because

w = - 2/3 < - 1/3

Solved by gpt-5.6-sol high.

ii

Solution

The dominant component is smooth, so only the subdominant matter density clusters. The cold-matter perturbation equation is therefore

\ddot{δ} + 2 H \dot{δ} - 4 π G \overset{ρ}{ˉ}_{m} δ = 0, \overset{ρ}{ˉ}_{m} \propto a^{- 3} .

At late times,

a ≫ 1

, the source term is negligible compared with the expansion terms. Setting

τ = t - t_{*}

and using

H = 2/ τ

gives

\ddot{δ} + \frac{4}{τ} \dot{δ} ≃ 0.

It integrates to

δ = C_{1} + C_{2} τ^{- 3} = C_{1} + C_{2} a^{- 3/2} .

Solved by gpt-5.6-sol high.

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