= Solution
First consider $k\tau_{\rm eq}<1$. 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>:
$$
P(k,\tau)\sim\mathcal C\tau^4 k,\qquad k\tau_{\rm eq}<1,
$$
up to a scale-independent matching coefficient.
For $k\tau_{\rm eq}\gg1$, 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. Thus
$$
\delta_{\rm eq}\sim\delta_H\ln(k\tau_{\rm eq}),\qquad
P_{\rm eq}\sim\mathcal C k^{-3}\ln^2(k\tau_{\rm eq}).
$$
During <matter domination>, the <matter-era linear growth factor> of the density is $(\tau/\tau_{\rm eq})^2$, so its square multiplies the <matter power spectrum>:
$$
\boxed{P(k,\tau)\sim\mathcal C
\left(\frac{\tau}{\tau_{\rm eq}}\right)^4
k^{-3}\ln^2(k\tau_{\rm eq}),\qquad k\tau_{\rm eq}\gg1.}
$$
The turnover at $k\sim\tau_{\rm eq}^{-1}$ 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 $k\tau_{\rm eq}=1$, where the spectrum does not vanish.
Multiplying by $k^3$ gives the small-scale mass-fluctuation <variance>
$$
\boxed{\left\langle\left(\frac{\delta M}{M}\right)_k^2\right\rangle
\sim\mathcal C\left(\frac{\tau}{\tau_{\rm eq}}\right)^4
\ln^2(k\tau_{\rm eq}).}
$$
This 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 requires
$$
\boxed{\mathcal C\sim
\frac{(\tau_{\rm eq}/\tau_0)^4}{\ln^2(\tau_{\rm eq}/L)}.}
$$
For a numerical illustration faithful to this question's flat, matter-plus-radiation background, take $h=0.7$ and $\Omega_r=4.2\times10^{-5}h^{-2}$, hence $\Omega_C=1-\Omega_r\simeq1$ rather than the $0.3$ appropriate to a vacuum-dominated model. Part 1(b), now with $\Omega_m=\Omega_C$, gives $\tau_{\rm eq}\simeq33\,\mathrm{Mpc}$. In this two-component model the same integral is valid all the way to today, giving
$$
\tau_0=\frac{2}{H_0\Omega_C}
\left(\sqrt{\Omega_r+\Omega_C}-\sqrt{\Omega_r}\right)
\simeq8.5\times10^3\,\mathrm{Mpc}.
$$
With $L=3\,\mathrm{Mpc}$, $\tau_0/\tau_{\rm eq}\simeq258$ and $\ln(\tau_{\rm eq}/L)\simeq2.39$. Therefore $\mathcal C\sim[258^4(2.39)^2]^{-1}\simeq4\times10^{-11}$, corresponding to a horizon-entry root-mean-square fluctuation of roughly $6\times10^{-6}$. \b[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.
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