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The broad turnover near $k\sim10^{-2}\,\mathrm{Mpc}^{-1}$ 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 as
$$
\Delta_m(k,a)=\frac25\frac{k^2}{\Omega_{m0}H_0^2}
T(k)D(a)\mathcal R(k),
$$
up to an irrelevant overall sign convention for $\mathcal R$. Normalize the <cosmological transfer function> by $T\to1$ 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 $k^{-3}\mathcal P_{\mathcal R}(k)$. Therefore
$$
P_m(k,a)\propto D(a)^2k\,T(k)^2\mathcal P_{\mathcal R}(k).
$$
For locally constant <scalar spectral index> $n_s$, the <asymptotic cold-dark-matter power spectrum> is
$$
\boxed{P_m(k)\propto
\begin{cases}
k^{n_s},&k\ll k_{\rm eq},\\
k^{n_s-4}\ln^2(k/k_{\rm eq}),&k\gg k_{\rm eq},
\end{cases}}
$$
because the <cold-dark-matter transfer function> tends to one at small $k$ and scales as $\ln(k/k_{\rm eq})/(k/k_{\rm eq})^2$ at large $k$. The asymptotes refer to the smooth envelope, not the individual baryonic wiggles. The large-$k$ local logarithmic slope is $n_s-4+2/\ln(k/k_{\rm eq})$. For the quadratic potential at $N\simeq60$, derived below, $n_s\simeq0.967$, 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.
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