Dimensional and dimensionless matter spectra across equality ($\mathcal{P}(k)=k^{3}P(k)/(2{\pi }^2)$)

The dimensionless spectrum is $\mathcal{P}(k)=k^{3}P(k)/(2{\pi }^2)$. If it rises as $k^4$ below equality and as ${\ln }^{2}k$ above equality, then the dimensional spectrum rises as $k$ and falls as $k^{-3}{\ln }^{2}k$. Thus the dimensional spectrum has a turnover while the dimensionless spectrum only bends to a much slower rise. A schematic transition must smoothly join the two asymptotic regimes.