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Take the regular growing matter mode ${\delta }_c=C{\tau }^2$ and $h=-2C{\tau }^2$. With $\mathcal{H}=2/\tau$, the constraint givesConsequentlyHere ${\zeta }_{\ast }=10C/k^2$ is its early, superhorizon constant value. The fractional time variation is of second gradient order, andThus the primordial curvature is conserved to leading order on superhorizon scales. This is the matter-era density transfer from primordial curvature.

A mode entering after equality evolves in the growing matter solution, givingThis assumes matter-dominated growth to ${\tau }_0$, as in the model. A subsequent dark-energy era would multiply by the appropriate late-time growth suppression. The initial radiation-era adiabatic curvature passes through equality unchanged at leading superhorizon order, so the same ${\zeta }_{\ast }$ is the primordial input.

For $k\gg k_{\mathrm{eq}}$, entry occurs in radiation domination at ${\tau }_h\sim k^{-1}$. The entry contrast is of order ${\zeta }_{\ast }$ because its superhorizon value scales as $k^2{\tau }_h^2{\zeta }_{\ast }$. In the question's stagnation approximation it stays of this order until equality, then grows by $({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2$. Thusindependent of $k$ to that approximation. The precise coefficient needs matching through the equality transition; the horizon convention, including the use of wavelength $2\pi /k$, also changes the matching definition of $k_{\mathrm{eq}}\sim {\tau }_{\mathrm{eq}}^{-1}$ by a fixed factor.

One continuous sharp-equality model, normalizing the plateau by matching the low-$k$ branch at $k_{\mathrm{eq}}$, is the sharp-equality cold-dark-matter transfer approximationThis is a matching approximation, not an exact solution through equality. The function requested here transfers curvature directly to density. It differs from the conventional normalized matter transfer $\mathcal{T}(k)$, where one factors out $k^2{\tau }_0^2/10$: that normalized function tends to one at low $k$ and to $(k_{\mathrm{eq}}/k{)}^2$ in the stagnation model at high $k$.

For the dimensional primordial spectrum $P_{\zeta }=A/k^3$, linear deterministic transfer gives $P_c=|T_{\delta }{|}^2{P}_{\zeta }$. Hence this sharp model yieldsThe spectrum turns over near equality. Retaining the slow radiation-era logarithmic growth refines the high-$k$ transfer to $T_{\delta }\propto ({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2\ln (k/k_{\mathrm{eq}})$, up to constants in the logarithm and matching factors, and therefore refines the small-scale spectrum to $P_c\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$. This is the logarithmic small-scale matter transfer correction to the question's constant-growth idealization. Multiplication by $k^3/(2{\pi }^2)$ would give dimensionless power, whose large-scale slope is $k^4$ rather than the dimensional spectrum's $k$.