sigma8

Sigma8, often written as sigma8, is a cosmological parameter that quantifies the amplitude of matter density fluctuations on scales of 8 h^-1 megaparsecs. It is defined as the root-mean-square of the linear matter overdensity after smoothing the density field with a spherical top-hat filter of radius R = 8 h^-1 Mpc, evaluated at the present epoch (z = 0). In terms of the matter power spectrum P(k), sigma_R^2 = (1/2π^2) ∫ P(k) W^2(kR) k^2 dk, where W is the Fourier transform of the top-hat window. Thus sigma8 = sigma_R with R = 8 h^-1 Mpc. In linear theory, sigma8 evolves with redshift according to the linear growth factor D(z), so sigma8(z) = sigma8(0) D(z)/D(0).

Observationally, sigma8 is constrained by the cosmic microwave background, galaxy clustering, weak gravitational lensing, cluster counts, and the Lyman-alpha forest. The central values are typically around 0.8, with uncertainties that depend on the data and modeling assumptions. There is a mild tension between some high-redshift CMB inferences and low-redshift probes, often discussed in terms of the derived S8 parameter, defined as S8 = sigma8 sqrt(Omega_m/0.3).

In cosmological analyses, sigma8 is degenerate with the matter density Omega_m and with other parameters that set the amplitude and growth of structure. Combined datasets are used to break these degeneracies and to constrain the overall normalization of the matter power spectrum, the abundance of massive halos, and the resulting lensing signals.