= Solution
In <matter domination>, the <Einstein field equations> with constant $\Phi$ give the <Newtonian-gauge matter density from a constant gravitational potential>
$$
-k^2\Phi-3\mathcal H^2\Phi=4\pi Ga^2\bar\rho_c\delta_c
=\frac32\mathcal H^2\delta_c,
\qquad
\boxed{\delta_c=-2\Phi-\frac{2k^2}{3\mathcal H^2}\Phi}.
$$
Well inside the <Hubble radius>, the first term is negligible. Since $\mathcal H^2\propto a^{-1}$ and $\Phi$ is constant,
$$
\boxed{\delta_{c,\rm grow}\propto a}.
$$
Equivalently, the <matter-era growing and decaying density modes> follow from $\delta_c''+\mathcal H\delta_c'-(3/2)\mathcal H^2\delta_c=0$: they are $\tau^2\propto a$ and $\tau^{-3}\propto a^{-3/2}$.
The lower panel of the preceding figure gives the three requested <density contrast> sketches in <Newtonian gauge in cosmology>. Their early superhorizon density is nearly constant, rather than proportional to $a^2$ in this gauge. A large-$k$ mode enters first and grows only approximately logarithmically during <radiation domination>. Once the rapid radiation forcing has subsided, $\delta_c''+\mathcal H\delta_c'\simeq0$ with $\mathcal H\simeq1/\tau$, so $\delta_c\simeq C+D\ln\tau\simeq C+D\ln a$. This is the <Mészáros effect>; forcing around entry determines the coefficients. After equality its growing part becomes proportional to $a$.
A mode near $k_{\rm eq}$ begins substantial growth around equality. A small-$k$ mode keeps its superhorizon constant term until entering during <matter domination>, then follows the same $a$ growth law. Their entry scale factors satisfy $a_{\rm ent}\propto k^{-1}$ during <radiation domination> and $a_{\rm ent}\propto k^{-2}$ during <matter domination>. At a common late time, the scaled amplitudes are therefore of order
$$
k^{3/2}\delta_c\propto
\begin{cases}
a\,k^2,&\mathcal H(a)\ll k\ll k_{\rm eq},\\
a\,\ln(k/k_{\rm eq}),&k\gg k_{\rm eq},
\end{cases}
$$
up to common dimensional constants and order-one matching terms. Thus the three late growth curves have the same logarithmic slope one as functions of $a$, but different amplitudes. These amplitude statements require the modes to have entered the <Hubble radius>; the full Newtonian-gauge formula above remains available for modes that have not.
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