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First consider $k{\tau }_{\mathrm{eq}}<1$. Such a mode remains outside the horizon through matter-radiation equality, grows as ${\tau }^2$ on both sides, and keeps the same matter-era growth after entry. Squaring this growth preserves the shape of the HPZ spectrum:up to a scale-independent matching coefficient.

For $k{\tau }_{\mathrm{eq}}\gg 1$, entry occurs in radiation domination at ${\tau }_H\sim k^{-1}$, with $P_H\sim \mathcal{C}{k}^{-3}$. The density amplitude then grows only logarithmically until equality. ThusDuring matter domination, the matter-era linear growth factor of the density is $(\tau /{\tau }_{\mathrm{eq}}{)}^2$, so its square multiplies the matter power spectrum:The turnover at $k\sim {\tau }_{\mathrm{eq}}^{-1}$ records whether a mode entered during radiation domination. Earlier-entering small-scale modes miss the rapid ${\tau }^2$ growth and receive only the logarithmic Mészáros effect. This is the origin of the logarithmic small-scale matter transfer. The formula is a large-$k$ asymptote: the additive constant in the logarithmic solution and smooth equality matching must be retained near $k{\tau }_{\mathrm{eq}}=1$, where the spectrum does not vanish.

Multiplying by $k^3$ gives the small-scale mass-fluctuation varianceThis gives the requested estimate for modes entering before equality, with the logarithm replaced by a smooth order-one factor in the transition region. On comoving scale $L\sim k^{-1}$, unit variance today requiresFor a numerical illustration faithful to this question's flat, matter-plus-radiation background, take $h=0.7$ and ${\Omega }_r=4.2\times 10^{-5}{h}^{-2}$, hence ${\Omega }_C=1-{\Omega }_r\simeq 1$ rather than the $0.3$ appropriate to a vacuum-dominated model. Part 1(b), now with ${\Omega }_m={\Omega }_C$, gives ${\tau }_{\mathrm{eq}}\simeq 33\mathrm{Mpc}$. In this two-component model the same integral is valid all the way to today, givingWith $L=3\mathrm{Mpc}$, ${\tau }_0/{\tau }_{\mathrm{eq}}\simeq 258$ and $\ln ({\tau }_{\mathrm{eq}}/L)\simeq 2.39$. Therefore $\mathcal{C}\sim [258^4(2.39{)}^2{]}^{-1}\simeq 4\times 10^{-11}$, corresponding to a horizon-entry root-mean-square fluctuation of roughly $6\times 10^{-6}$. A very small primordial amplitude, of order $10^{-10}$ in this schematic normalization, can reach unit small-scale variance by today. Factors from the Fourier convention, smoothing window and equality matching preclude a precise normalization here. The calculation marks the onset of nonlinear structure formation, where extrapolating the linear solution ceases to be valid. For a universe with late vacuum domination, one must replace the matter-only growth law by its actual growth factor before making a numerical inference.

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 $k$.

The Mészáros effect suppresses the growth of modes which enter the horizon during radiation domination. Modes with comoving wavenumber $k\gg k_{\mathrm{eq}}$ enter earlier, but gain only a logarithmic growth factor until matter-radiation equality; modes entering during matter domination grow as $a$ after entry. Consequently small-scale amplitudes are reduced relative to an extrapolation of the unsuppressed large-scale cosmological density power spectrum, creating a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}={\mathcal{H}}_{\mathrm{eq}}$ in units $c=1$.

More quantitatively, for adiabatic primordial fluctuations with scalar spectral index $n_s$, the late-time linear spectrum can be written, up to a time-dependent normalization,The cold-dark-matter transfer function approaches a constant at $k\ll k_{\mathrm{eq}}$, whereas logarithmic small-scale matter transfer gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\log (k/k_{\mathrm{eq}})$ at large $k$. ThusFor approximately scale-invariant initial curvature fluctuations, $n_s\simeq 1$, the large-$k$ scaling is $k^{-3}{\log }^{2}k$, rather than $k$. This discussion ignores baryonic acoustic structure and nonlinear clustering. The Mészáros equation solution basis includes both a constant and a logarithmic branch during radiation domination; the physical logarithmic growth is selected by matching the earlier radiation forcing, so not every mathematical solution grows logarithmically.