Asymptotic cold-dark-matter power spectrum ($P_m\propto k^{n_s}T(k{)}^2$)

The dimensional matter power spectrum contains a factor $k^{n_s}$, rather than $k^{n_s+3}$. The cosmological transfer function tends to one below the matter-radiation equality scale and to $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ above it. The smooth spectrum therefore scales as $k^{n_s}$ at small wavenumber and $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$ at large wavenumber. The logarithmic factor records slow cold-matter growth during radiation domination. Multiplying by $k^3$ instead produces dimensionless matter power.