Logarithmic high-wavenumber density-spectrum transfer (${\mathcal{P}}_c\propto (\tau /{\tau }_{\mathrm{eq}}{)}^4{\ln }^2(k{\tau }_{\mathrm{eq}})$)

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.