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The plot uses comoving wavenumber in $\mathrm{Mp}{\mathrm{c}}^{-1}$. Using the inverse comoving particle horizon from part (ii) gives $1/r_{p,\mathrm{eq}}\simeq 9\times 10^{-3}\mathrm{Mp}{\mathrm{c}}^{-1}$. The conventional matter-radiation equality scale is insteadThese are consistent order-one conventions for locating the turnover near $10^{-2}\mathrm{Mp}{\mathrm{c}}^{-1}$. The horizon-entry condition is $k/(aH/c)\sim 1$; equating an entire Fourier wavelength $2\pi /k$ to the particle horizon would attach a different numerical factor.

For adiabatic initial conditions, the asymptotic cold-dark-matter power spectrum has $P(k)\propto k^{n_s}T(k{)}^2$, where the scalar spectral index is $n_s\simeq 1$ and $T$ is the cosmological transfer function. Modes below $k_{\mathrm{eq}}$ enter during matter domination and have little relative suppression, giving approximately $P\propto k$. Modes above $k_{\mathrm{eq}}$ enter during radiation domination, when logarithmic growth of matter perturbations during radiation domination is much slower. Their transfer is approximately $T\propto \ln (k/k_{\mathrm{eq}})/k^2$, giving $P\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$ well above equality. This explains the broad turnover; baryon acoustic oscillations add structure, and nonlinear evolution modifies sufficiently small scales.

The broad turnover near $k\sim 10^{-2}{\mathrm{Mpc}}^{-1}$ marks the matter-radiation equality scale. Modes entering the Hubble radius after matter-radiation equality grow efficiently during matter domination. Modes entering earlier have their cold-matter growth delayed by radiation domination: their subhorizon growth is logarithmic rather than proportional to $a$. This is the Mészáros effect. The smaller undulations on the declining side are baryon acoustic oscillations inherited from the photon-baryon fluid, with an oscillatory wavenumber spacing set approximately by $\pi$ divided by the sound horizon. They are damped at sufficiently short scales by photon diffusion and the transfer of the baryon perturbation into the total matter field. The plotted spectrum is a linear prediction; nonlinear evolution changes the actual sufficiently small-scale spectrum at redshift zero.

For an adiabatic growing mode, the comoving curvature perturbation transfers to the matter overdensity asup to an irrelevant overall sign convention for $\mathcal{R}$. Normalize the cosmological transfer function by $T\to 1$ at small $k$ and the linear growth factor by $D\to a$ in matter domination. The relation distinguishes a dimensional matter power spectrum from a dimensionless cosmological power spectrum: the dimensional curvature spectrum is proportional to $k^{-3}{\mathcal{P}}_{\mathcal{R}}(k)$. ThereforeFor locally constant scalar spectral index $n_s$, the asymptotic cold-dark-matter power spectrum isbecause the cold-dark-matter transfer function tends to one at small $k$ and scales as $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. The asymptotes refer to the smooth envelope, not the individual baryonic wiggles. The large-$k$ local logarithmic slope is $n_s-4+2/\ln (k/k_{\mathrm{eq}})$. For the quadratic potential at $N\simeq 60$, derived below, $n_s\simeq 0.967$, so the limiting powers are approximately $k^{0.967}$ and $k^{-3.033}{\ln }^{2}k$. Multiplying by $k^3$ to form dimensionless matter power would change both exponents by three; that is not the plotted quantity.