= Solution
For a fixed coordinate cell, the spatial volume element is
$$
\sqrt{\det g_{ij}}\,d^3x=a^3\sqrt{\det(\delta_{ij}+h_{ij})}\,d^3x
=a^3(1+\tfrac12h)d^3x+O(h^2).
$$
Thus its perturbation relative to the background expansion, or its perturbed <comoving volume>, is $\Delta V_c/V_c=h/2$. In the cold-matter rest frame each such cell contains a fixed amount of matter. Since density is mass divided by physical volume,
$$
\boxed{\delta_c=-\frac{\Delta V_c}{V_c}=-\frac h2.}
$$
Equivalently, <stress-energy conservation> gives $\delta_c'=-h'/2$; the stated relation fixes the spatially dependent integration constant by the initial mass-coordinate convention. A contraction of <comoving volume> gives a positive <density contrast>.
In the matter era $a\propto\tau^2$, so $\mathcal H=2/\tau$ and the matter density fraction is approximately one. The evolution equation reduces to
$$
\delta_c''+\frac2\tau\delta_c'-\frac6{\tau^2}\delta_c=0.
$$
For $\delta_c\propto\tau^p$, the indicial equation is $p(p-1)+2p-6=(p-2)(p+3)=0$. Therefore
$$
\boxed{\delta_c(\mathbf k,\tau)=C_+(\mathbf k)\tau^2+C_-(\mathbf k)\tau^{-3}.}
$$
The growing mode is proportional to $a$, and the decaying mode to $a^{-3/2}$. These are the density solutions after the comoving coordinate choice; the residual time mode found in part (i) explains why a similar power in an unfixed density variable cannot be interpreted in isolation.
There is a normalization omission in the printed density parameter if $\bar\rho_c$ denotes physical density. Its usual definition in <conformal time> is
$$
\Omega_c=\frac{8\pi G\bar\rho_c}{3H^2}=\frac{8\pi G a^2\bar\rho_c}{3\mathcal H^2},\qquad H=\frac{a'}{a^2}.
$$
The printed expression lacks $a^2$, unless its density is already rescaled by that factor. The matter-era statement $\Omega_c\simeq1$ and the growth equation use the physical density fraction above.
On <superhorizon scales> during <radiation domination>, the regular <adiabatic cosmological perturbation> has approximately constant primordial curvature. The comoving density perturbation is gradient-suppressed, of order $(k/\mathcal H)^2$ times that curvature. Since $\mathcal H\propto\tau^{-1}$, it grows as $\delta_c\propto k^2\tau^2$ for fixed mode amplitude. This is the regular superhorizon solution in the specified slicing, not causal collapse driven by communication across the entire wavelength.
After radiation-era horizon entry, the dominant radiation develops pressure-supported acoustic oscillations. Its potential no longer provides a steady growing gravitational source, while cold matter is too small a fraction of the total density to produce rapid self-gravitating growth. The result is the <Mészáros effect>: growth is strongly suppressed until equality. The supplied stagnation approximation replaces that slow evolution by a constant. More accurately, ignoring the rapidly oscillating source at leading order gives $\delta_c''+\delta_c'/\tau\simeq0$, with a constant and a logarithmic solution. Thus \b[radiation-era subhorizon growth is at most slow logarithmic growth, rather than the matter-era power law]; treating it as constant is the explicit approximation used for the simple transfer estimate below.
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