Eight-megaparsec density fluctuation amplitude (${\sigma }_8$)

The parameter ${\sigma }_8$ is the present linear rms density contrast averaged in a spherical top-hat of comoving radius $8h^{-1}\mathrm{Mpc}$, with $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})$. Thus ${\sigma }_8^2=(2{\pi }^2{)}^{-1}{\int }_0^{\infty }{k}^2{P}_m(k,0)|W(k{R}_8){|}^{2}dk$. It specifies an amplitude on a chosen scale rather than the complete matter power spectrum; converting cluster abundance into this parameter also requires a spectral shape and background cosmology.