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Let $\delta (\mathbf{x},z)$ be the linear density contrast and ${\delta }_R$ its convolution with a normalized smoothing window function. For a statistically homogeneous and isotropic field with matter power spectrum $P(k,z)$,The variance is ${\sigma }^2$; the root-mean-square fluctuation is its square root $\sigma$. The formula needs a convergent integral and does not require the field to be Gaussian.

For a spherical top-hat window function of comoving radius $R$, the Lagrangian mass label is $M=4\pi {\bar{\rho }}_{m,0}{R}^3/3$, where ${\bar{\rho }}_{m,0}$ is the present mean matter mass density and $a_0=1$. One therefore writes the same quantity as ${\sigma }^2(M,z)$. This is the initially assigned mass scale, rather than the final physical radius of a collapsed halo. For scale-independent linear growth factor $D(z)$, $\sigma (M,z)$ scales as $D(z)$ at fixed $M$.

A collapse calculation must use a compatible smoothing window, epoch and barrier convention. Large extrapolated linear variance indicates that nonlinear evolution matters; it does not turn the linear spectrum into the true nonlinear density distribution. The eight-megaparsec density fluctuation amplitude ${\sigma }_8$ is one particular present-day top-hat rms value.