For matter plus a cosmological constant,
$$
H^2=H_0^2(\Omega_{m,0}a^{-3}+\Omega_{\Lambda,0}).
$$
Set $\delta_m=Hu$. Substitution into the equation from part (c) makes the coefficient of $u$ vanish by the Friedmann relation and leaves
$$
u''+\left(3\frac{H'}H+\frac3a\right)u'=0,
$$
or
$$
\frac d{da}(a^3H^3u')=0.
$$
Therefore the two independent modes can be written
$$
\delta_m=C_1H(a)+C_2H(a)
\int^a\frac{da'}{a'^3H(a')^3}.
$$
The first is the decaying mode. Normalizing the second to $D_+(a)\sim a$ at early times gives the <integral linear growth factor in a matter-Lambda universe>
$$
\boxed{D_+(a)=\frac52\Omega_{m,0}H_0^2H(a)
\int_0^a\frac{da'}{a'^3H(a')^3}}.
$$

Solution

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

{title2=$D_+(a)$}

The growing solution of the matter perturbation equation in a spatially flat matter-plus-cosmological-constant universe is
$$
D_+(a)=\frac52\Omega_{m,0}H_0^2H(a)
\int_0^a\frac{da'}{a'^3H(a')^3},
$$
normalized so that $D_+(a)\sim a$ during early matter domination.


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