I'm writing the introduction for my thesis and reading the Book, 'Galaxy Formation and Evolution' by Mo et al. There is a clear and accessible explanation of what the σ_8 parameter really means:

In the linear regime the density field

In order to fully specify the density field in the early Universe we have to observationally constrain the amplitude of the H-Z spectrum. This amplitude sets the amount of clustering of structure, frequently estimated from the two-point correlation function.

When someone measures σ_8 from galaxy clustering now it means that he or she has calculated the variance (difference from the background density) of galaxy number counts within randomly placed spheres with radius R = 8h^-1 Mpc. For galaxies of Milky Way luminosity the variance is ~= 1, however, such variance decreases sharply if the radii of such spheres approach 30h^-1 Mpc. This means that the Universe is non-homogeneous on small scales, but approaches homogeneity on larger ones.

Note that the cosmological parameter σ_8 and the observed galaxy number variance in spheres of radius 8h^-1 Mpc are not equivalent. The structures we observe today are very far in the non-linear regime.

In the linear regime the density field

*ρ*is Gaussian, characterised by the scale-invariant Harrison-Zel'dovich spectrum. It means that the power of a mode*P(k) ∝ k^(n - 1)*, and n = 1. This ensures that the gravitational potential does not diverge neither on small nor on large scales.In order to fully specify the density field in the early Universe we have to observationally constrain the amplitude of the H-Z spectrum. This amplitude sets the amount of clustering of structure, frequently estimated from the two-point correlation function.

When someone measures σ_8 from galaxy clustering now it means that he or she has calculated the variance (difference from the background density) of galaxy number counts within randomly placed spheres with radius R = 8h^-1 Mpc. For galaxies of Milky Way luminosity the variance is ~= 1, however, such variance decreases sharply if the radii of such spheres approach 30h^-1 Mpc. This means that the Universe is non-homogeneous on small scales, but approaches homogeneity on larger ones.

Note that the cosmological parameter σ_8 and the observed galaxy number variance in spheres of radius 8h^-1 Mpc are not equivalent. The structures we observe today are very far in the non-linear regime.

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