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There is a conformal-time normalization error in the printed density parameter. For physical background density, $H=\mathcal{H}/a$, so the cosmological density parameter isThe printed definition lacks the numerator's $a^2$. The scale factor in a conformal-time density parameter is essential for ${\Omega }_c\simeq 1$ in the intended flat matter-dominated approximation.

During matter domination, pressureless matter supplies both the expansion and the gravitational source: $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$ and ${\Omega }_c\simeq 1$ when CDM is taken to constitute that matter. ThusTry ${\delta }_c\propto {\tau }^p$. Its characteristic polynomial is $p(p-1)+2p-6=(p-2)(p+3)$, givingRadiation-era photon pressure and radiation perturbations affect the metric source, and matter alone does not dominate the expansion, so this matter-only closure and its power laws are inadequate before equality. At late acceleration the growth also ceases to follow $a\propto {\tau }^2$ and slows. The question's claim that the differential form itself is valid only in matter domination is too strong: for subhorizon pressureless matter with smooth dark energy it still applies with the correct time-dependent $\mathcal{H}$ and ${\Omega }_m$. Indeed, the pressureless growth equation in proper time is $\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$ for smooth dark energy; substituting $d/d{t}_{\mathrm{proper}}=a^{-1}d/d\tau$ gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }-4\pi G{a}^2{\bar{\rho }}_m\delta =0$, the same conformal differential form. It is the matter-dominated coefficients and simple solutions that are restricted to the stated interval.

Outside the radiation-era horizon, the regular growing adiabatic mode has nearly constant curvature, while its comoving-synchronous density perturbation begins at gradient order $k^2{\tau }^2$. Equivalently its metric trace grows as ${\tau }^2$ and the volume relation in (i) gives ${\delta }_c\propto {\tau }^2$. This is a relativistic superhorizon solution, not instantaneous Newtonian communication between causally disconnected regions. Inside the horizon, pressure makes the dominant radiation oscillate and its gravitational potential decay, while CDM self-gravity is still subdominant. CDM then grows only weakly, logarithmically in the more accurate Mészáros effect; the stated freezing approximation replaces that slow growth by a constant.

Assume all modes being compared start outside the horizon at common ${\tau }_i$, and define ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and $G_0=({\tau }_0/{\tau }_i{)}^2$. Modes with $k\le k_{\mathrm{eq}}$ never undergo the radiation-era frozen interval, so their growth is $G_0$. For $k>k_{\mathrm{eq}}$, multiply growth before horizon entry, freezing until equality, and subsequent matter growth:Thus the frozen-radiation approximation to the matter transfer function isThis $T$ includes total growth because the question defines it directly between two density contrasts. The usual transfer normalized to one on large scales is $T/G_0$.

Power is the variance of a Fourier amplitude, so $P_0=|T{|}^2{P}_i$. The prescribed primordial density spectrum givesIt turns over at the comoving equality-horizon scale $2\pi /k_{\mathrm{eq}}={\tau }_{\mathrm{eq}}$. The turnover and reduced small-scale growth help organize the broad spectrum of galaxy clustering, clusters and superclusters; they do not select one unique object size. The dimensionless power $k^3{P}_0/(2{\pi }^2)$ is approximately flat above equality in this crude approximation, so a decreasing dimensional spectrum does not imply no small-scale structures. Restoring radiation-era logarithmic growth adds logarithmic corrections to the small-scale tail. Baryon acoustic features, a smooth equality transition, late growth suppression and nonlinear halo formation are absent from this simple linear broken-power approximation.