= Solution
The plot uses comoving <wavenumber> in $\mathrm{Mpc^{-1}}$. Using the inverse <comoving particle horizon> from part (ii) gives $1/r_{p,\rm eq}\simeq9\times10^{-3}\,\mathrm{Mpc^{-1}}$. The conventional <matter-radiation equality scale> is instead
$$
k_{\rm eq}=\frac{a_{\rm eq}H_{\rm eq}}c\simeq\frac{H_0}c\sqrt{\frac{2\Omega_{m,0}}{a_{\rm eq}}}\simeq\boxed{1.1\times10^{-2}\,\mathrm{Mpc^{-1}}.}
$$
These are consistent order-one conventions for locating the turnover near $10^{-2}\,\mathrm{Mpc^{-1}}$. The horizon-entry condition is $k/(aH/c)\sim1$; 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 $n_s\simeq1$ and $T$ is the <cosmological transfer function>. Modes below $k_{\rm eq}$ enter during <matter domination> and have little relative suppression, giving approximately $P\propto k$. Modes above $k_{\rm eq}$ enter during <radiation domination>, when <logarithmic growth of matter perturbations during radiation domination> is much slower. Their transfer is approximately $T\propto\ln(k/k_{\rm eq})/k^2$, giving $P\propto k^{-3}\ln^2(k/k_{\rm eq})$ well above equality. This explains the broad turnover; <baryon acoustic oscillations> add structure, and nonlinear evolution modifies sufficiently small scales.
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