Logarithmic small-scale matter transfer ($T(k)\propto \log (k/k_{\mathrm{eq}})(k_{\mathrm{eq}}/k{)}^2$)

Modes with $k\gg k_{\mathrm{eq}}$ 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 $P(k)\propto k^{n_s-4}{\log }^2(k/k_{\mathrm{eq}})$ at large $k$. These scalings omit baryonic acoustic structure and nonlinear evolution.

Here the cold-dark-matter transfer function is normalized so that ${\delta }_m(k,a)=A(a)k^{2}T(k)\mathcal{R}(k)$, where $\mathcal{R}$ is the primordial comoving curvature perturbation and $A(a)$ is independent of $k$. During radiation domination, horizon entry occurs at $a_{\mathrm{ent}}\propto k^{-1}$. An adiabatic entry amplitude of order $\mathcal{R}$ then acquires a factor $\log (a_{\mathrm{eq}}/a_{\mathrm{ent}})\sim \log (k/k_{\mathrm{eq}})$ by matter-radiation equality. Subsequent matter domination supplies a common growth factor; division by the conventional $k^2$ prefactor therefore gives $T(k)\propto k^{-2}\log (k/k_{\mathrm{eq}})$.

A scale-invariant curvature seed gives wavenumber-independent dimensionless matter power at radiation-era horizon entry. Horizon entry at ${\tau }_h\sim 1/k$, followed by logarithmic growth to equality, gives a density transfer proportional to $\ln (k{\tau }_{\mathrm{eq}})$. Subsequent matter-era growth is proportional to ${\tau }^2$, so the dimensionless density power is proportional to $(\tau /{\tau }_{\mathrm{eq}}{)}^4{\ln }^2(k{\tau }_{\mathrm{eq}})$. This is a high-wavenumber asymptote, not an expression at the transition point.