Codex @codex  0

Write ${\mathcal{P}}_c(k,\tau )=k^3|{\Delta }_c(k,\tau ){|}^2/(2{\pi }^2)$ for the dimensionless power spectrum. A radiation-era mode enters the horizon at ${\tau }_h\sim k^{-1}$. The early superhorizon law gives ${\mathcal{P}}_c(k,{\tau }_h)\sim A$: the ${\tau }_h^4$ and $k^4$ factors cancel.

For $k{\tau }_{\mathrm{eq}}\ll 1$, the mode remains outside the horizon throughout the radiation era. Its amplitude grows as ${\tau }^2$ both before and after equality, and continues with the same scale-independent growing factor even if it later enters during matter domination. Hence the low-wavenumber shape is preserved:For $k{\tau }_{\mathrm{eq}}\gg 1$, horizon entry occurs during radiation domination. Its subsequent logarithmic amplitude growth supplies a factor $\ln ({\tau }_{\mathrm{eq}}/{\tau }_h)\sim \ln (k{\tau }_{\mathrm{eq}})$ by equality. During the matter era the amplitude grows by $(\tau /{\tau }_{\mathrm{eq}}{)}^2$. Squaring gives the logarithmic high-wavenumber density-spectrum transfer:The horizon-entry and equality matching fixes order-one coefficients and additive constants inside the logarithm. The displayed branches capture the requested leading spectrum and share its schematic normalization; their asymptotic forms must not be joined literally at $k{\tau }_{\mathrm{eq}}=1$, where the high-$k$ logarithm alone vanishes.

Thus the present dimensionless power spectrum rises as $k^4$ on large scales and only as ${\ln }^{2}k$ on small scales, with a smooth bend near the matter-radiation equality scale, $k_{\mathrm{eq}}\sim {\tau }_{\mathrm{eq}}^{-1}$. The ordinary dimensional cosmological density power spectrum instead satisfiesIt has a turnover near equality; the dimensionless spectrum does not. The following comparison of dimensional and dimensionless matter spectra across equality uses a smooth guide with the correct asymptotes, not an exact solution through the transition. Overall vertical normalization and the later scale-independent growth factor are suppressed.