= Logarithmic small-scale matter transfer
{title2=$T(k)\propto\log(k/k_{\rm eq})(k_{\rm eq}/k)^2$}
Modes with $k\gg k_{\rm eq}$ enter the horizon during <radiation domination>. Their matter perturbations grow only logarithmically until <matter-radiation equality>, rather than as $a$. Matching the radiation-era logarithmic growth onto the matter-era growing mode gives the large-$k$ <cosmological transfer function> scaling shown, up to constants and the matching scale inside the logarithm. For a primordial <scalar spectral index> $n_s$, the late-time linear <cosmological density power spectrum> consequently changes from $P(k)\propto k^{n_s}$ at small $k$ to $P(k)\propto k^{n_s-4}\log^2(k/k_{\rm eq})$ at large $k$. These scalings omit baryonic acoustic structure and nonlinear evolution.
Here the <cold-dark-matter transfer function> is normalized so that $\delta_m(k,a)=A(a)k^2T(k)\mathcal R(k)$, where $\mathcal R$ is the primordial <comoving curvature perturbation> and $A(a)$ is independent of $k$. During <radiation domination>, horizon entry occurs at $a_{\rm ent}\propto k^{-1}$. An adiabatic entry amplitude of order $\mathcal R$ then acquires a factor $\log(a_{\rm eq}/a_{\rm ent})\sim\log(k/k_{\rm eq})$ by <matter-radiation equality>. Subsequent <matter domination> supplies a common growth factor; division by the conventional $k^2$ prefactor therefore gives $T(k)\propto k^{-2}\log(k/k_{\rm eq})$.
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