massvariance

Mass variance, in cosmology and astrophysics, is a measure of the amplitude of fluctuations in the mass density field on a given length scale. It is defined as the variance of the smoothed density contrast, which quantifies how much the local mass density deviates from the cosmic mean when averaged over a region of size R.

Let δ(x) be the density contrast, δ(x) = (ρ(x) − ρ̄)/ρ̄, where ρ̄ is the mean density. After smoothing with a window function W_R, the smoothed density contrast is δ_R(x) = ∫ δ(y) W_R(|x − y|) d^3y. The mass variance on scale R is σ^2(R) = ⟨δ_R^2⟩, the ensemble average of the squared smoothed contrast. In Fourier space, this can be written as σ^2(R) = ∫ d^3k/(2π)^3 P(k) |W(kR)|^2, where P(k) is the matter power spectrum and W(kR) is the Fourier transform of the window. For a real-space spherical top-hat window of radius R, the enclosed mass is M = (4π/3) ρ̄ R^3, and σ^2(M) is often used interchangeably with σ^2(R).

Mass variance is a central input to models of structure formation, such as the Press–Schechter formalism, where the abundance of collapsed halos depends on σ(M) and its derivative. It encodes how fluctuations of different scales grow under gravity and thus influences the halo mass function. The shorthand σ8 denotes the root-mean-square fluctuation on 8 Mpc/h scales, a commonly used normalization. Variants include different window functions (e.g., Gaussian) that yield slightly different σ(R) for the same P(k).