How do you convert the value listed in Planck 2018 results. VI. Cosmological parameters, $A_s = 2.101\times10^{-9}$ to the value of the matter fluctuation amplitude $\sigma_8=0.8111$? I tried integrating this in with a growth function $D\sim 3400$, transfer function (using Eisenstein and Hu), and real-space tophat window function, but I'm always short of $0.8111$.
Also it seems that the wavenumber with regards to the power spectrum is not normalized by $2\pi$ so that $k_{\rm eq} \sim 0.01 \textrm{ Mpc}^{-1}$. Is it true then that the values given at $k_*=0.05 \textrm{ Mpc}^{-1}$ is within the horizon at matter-radiation equality, $A_s$ is not representative of the spectrum at large scales but somewhat suppressed?
